Approximation Limitations of Pure Dynamic Programming
aa r X i v : . [ c s . CC ] D ec APPROXIMATION LIMITATIONS OF PURE DYNAMIC PROGRAMMING
STASYS JUKNA AND HANNES SEIWERT A BSTRACT . We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+)and (max,+) circuits approximating given optimization problems. Many classical dynamic program-ming (DP) algorithms for optimization problems are pure in that they only use the basic min , max , + operations in their recursion equations. Tropical circuits constitute a rigorous mathematical model forthis class of algorithms. An algorithmic consequence of our lower bounds for tropical circuits is thatthe approximation powers of pure DP algorithms and greedy algorithms are incomparable. That pureDP algorithms can hardly beat greedy in approximation, is long known. New in this consequence is thatalso the converse holds. Keywords: dynamic programming, greedy algorithm, approximation, lower bounds
1. I
NTRODUCTION
A combinatorial optimization problem is specified by a finite set of ground elements and a family F of subsets of these elements, called feasible solutions . The problem itself then is, given an assignmentof nonnegative real weights to the ground elements, to compute the minimum or the maximum weightof a feasible solution, the latter being the sum of weights of its elements.The family F of feasible solutions itself can be described either explicitly, or as the set of - so-lutions of a system of linear inequalities (as in linear programming), or by other means. Important isonly that F does not depend on the actual input weighting: the family F is the same for all arrivinginput weightings.For example, in the MST problem (minimum weight spanning tree problem) on a given graph,feasible solutions are spanning trees of this graph (viewed as sets of their edges), and the problem isto compute the minimum weight of a spanning tree of this graph. In the assignment problem, feasiblesolutions are perfect matchings in a complete bipartite graph, etc.Dynamic programming (DP) is a fundamental algorithmic paradigm for solving combinatorial opti-mization problems. Many classical DP algorithms are pure in that they only apply the basic operations (min , +) or (max , +) in their recursion equations. Note that these are the only operations used in thedefinitions of the optimization problems themselves.Notable examples of pure DP algorithms for combinatorial optimization problems are the well-known Bellman–Ford–Moore shortest 𝑠 - 𝑡 path algorithm [4, 11, 22], the Floyd–Warshall all-pairsshortest paths algorithm [9, 29] (see Fig. 1), the Held–Karp traveling salesman algorithm [14], theDreyfus–Levin–Wagner Steiner tree algorithm [6, 19]. The Viterbi (max , ×) DP algorithm [28] is alsoa pure (min , +) DP algorithm via the isomorphism ℎ ∶ (0 , → ℝ + given by ℎ ( 𝑥 ) = − ln 𝑥 . This work was funded by the DFG (German Research Foundation) grant JU 3105/1-1. ∼ jukna/. Institute of Computer Science, Goethe University Frankfurt, Frankfurt am Main, Germany. Email: [email protected]. 𝑔 𝑘 −1 ( 𝑖, 𝑗 ) 𝑔 𝑘 −1 ( 𝑖, 𝑘 ) 𝑔 𝑘 −1 ( 𝑘, 𝑗 ) GFED@ABC min + 𝑔 𝑘 ( 𝑖, 𝑗 ) ❋❋❋❋❋❋❋❋❋❋❋❋ $ $ ■■■■■ z z ✉✉✉✉✉ z z ✉✉✉✉✉✉ F IGURE A fragment of a tropical (min , +) circuit of size 𝑂 ( 𝑛 ) im-plementing the Floyd–Warshall DP algorithm for the all-pairs lightestpaths on 𝐾 𝑛 problem. At the gate 𝑔 𝑘 ( 𝑖, 𝑗 ) , the minimum weight of a pathfrom 𝑖 to 𝑗 , which only uses nodes , … , 𝑘 as inner nodes, is computed. The main question we ask in this paper is: How many operations are necessary for pure DP algo-rithms to approximate a given combinatorial optimization problem within a given factor? That is, weare interested in proving lower bounds on the number of performed operations.A natural mathematical model for pure (min , +) and (max , +) DP algorithms is that of tropicalcircuits. A tropical (min , +) circuit is a directed acyclic graph, whose each indegree-zero node holdseither one of the input variables 𝑥 , … , 𝑥 𝑛 or a nonnegative real constant, and every other node (agate) has indegree two and computes either the minimum or the sum of the values computed at itstwo predecessors. Tropical (max , +) circuits are defined similarly. The size of a circuit is the totalnumber of its gates. Note that pure (min , +) and (max , +) DP algorithms are just special (recursivelyconstructed) tropical circuits (see Fig. 1). So, lower bounds on the size of tropical circuits show limitsof these pure DP algorithms.In this paper, we prove the first non-trivial, even super-polynomial, lower bounds for approximating tropical circuits and, hence, also for approximating pure DP algorithms.Recall that an algorithm approximates a given optimization problem 𝑓 within a factor 𝑟 ⩾ (or 𝑟 - approximates 𝑓 ) if for every input weighting 𝑥 (a vector of 𝑛 nonnegative real numbers), the outputvalue of the algorithm lies: ◦ between 𝑓 ( 𝑥 ) and 𝑟 ⋅ 𝑓 ( 𝑥 ) , in the case when 𝑓 is a minimization problem; ◦ between 𝑓 ( 𝑥 )∕ 𝑟 and 𝑓 ( 𝑥 ) , in the case when 𝑓 is a maximization problem.The factor 𝑟 may depend on the length 𝑛 of the inputs 𝑥 , but not on the inputs 𝑥 themselves. In bothcases, the smaller the factor 𝑟 is, the better is the approximation. In particular, factor 𝑟 = 1 means thatthe problem is solved exactly.One of our motivations for proving lower bounds on the number of operations performed by approx-imating pure DP algorithms is to compare their approximation power with that of the greedy algorithm;see Appendix A for what we mean by the greedy algorithm.That the greedy algorithm can have much worse approximation behavior than pure DP algorithmsis long known. Namely, there are many optimization problems easily solvable by pure DP algorithmsusing a small number of (min , +) or (max , +) operations, but the greedy algorithm cannot achieve anynon-trivial approximation factor (smaller than the maximum number of elements in feasible solutions).Such are, for example, the maximum weight independent set in a tree, or the maximum weight simplepath in a transitive tournament problem, and many other problems. To give a trivial example, notethat the problem 𝑓 ( 𝑥 ) = max{ 𝑥 , 𝑥 + ⋯ + 𝑥 𝑛 } can be solved (within factor 𝑟 = 1 ) by a trivial pure (max , +) DP algorithm performing only 𝑛 − 1 operations, but the greedy algorithm cannot achieve anysmaller than 𝑟 = 𝑛 − 1 approximation factor for this problem (see Proposition A.1 in Appendix A). Butwhat about the converse direction: can also pure DP algorithms have worse approximation behaviorthan greedy?Apparently, the first indication that greedy can also beat pure DP was given by Jerrum and Snir [15].They proved that every (min , +) circuit solving (exactly, within factor 𝑟 = 1 ) the directed MST problemon 𝑛 -vertex graphs (known also as the arborescence problem) requires Ω( 𝑛 ) gates. Since the family offeasible solutions of the arborescence problem is an intersection of two matroids, the greedy algorithm can approximate this problem within factor 𝑟 = 2 . This result was later improved in [17] by showingthat also the undirected MST problem, which can already be solved by the greedy algorithm exactly,requires (min , +) circuits of size Ω( √ 𝑛 ) to be solved exactly.But what if pure DP algorithms are only required to approximate a given optimization problemwithin some factor 𝑟 > ? Can greedy algorithms achieve smaller approximation factors than efficientpure DP algorithms? Our lower bounds on the size of approximating tropical circuits answer thisquestion in the affirmative .Below we summarize our main results. Since the approximation behaviors of tropical (min , +) and (max , +) circuits turned out to be completely different, we consider minimization and maximizationproblems separately. 2. M AIN RESULTS
Recall that a combinatorial optimization problem 𝑓 ( 𝑥 , … , 𝑥 𝑛 ) is specified by giving some family F ⊆ [ 𝑛 ] of feasible solutions. The problem itself is then, given an input weighting 𝑥 ∈ ℝ 𝑛 + , to computeeither the minimum or the maximum weight ∑ 𝑖 ∈ 𝑆 𝑥 𝑖 of a feasible solution 𝑆 ∈ F . To indicate thetotal number 𝑛 of ground elements, we will also write 𝑓 𝑛 instead of just 𝑓 . Minimization.
The boolean version of a minimization problem 𝑓 𝑛 is the monotone boolean functionwhich, given a set of ground elements, decides whether this set contains at least one feasible solutionof 𝑓 𝑛 . Result (min , +) circuits; Theorem 4.3) . If the boolean version of a minimizationproblem 𝑓 𝑛 requires monotone boolean (∨ , ∧) circuits of size > 𝑡 , then no tropical (min , +) circuit ofsize ⩽ 𝑡 can approximate 𝑓 𝑛 within any finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ .That is, if a tropical (min , +) circuit has fewer than 𝑡 gates, then regardless of how large approxima-tion factor 𝑟 we will allow, there will be an input weighting on which the circuit makes an error: thecomputed value on this input will be either strictly smaller or more than 𝑟 times larger than the optimalvalue.Together with known lower bounds for monotone boolean circuits, Result 1 yields the same lowerbounds for tropical (min , +) circuits approximating the corresponding minimization problems.Take, for example the assignment problem: given a nonnegative weighting of the edges of thecomplete bipartite 𝑛 × 𝑛 graph, compute the minimum weight of a perfect matching. Jerrum andSnir [15] have proved that any (min , +) circuit solving this problem exactly (within the factor 𝑟 = 1 )must have Ω( 𝑛 ) gates. On the other hand, together with Razborov’s monotone circuit lower bound forthe logical permanent function [26], Result 1 implies that a polynomial in 𝑛 number of gates is notsufficient to approximate this problem even when an arbitrarily large approximation factor is allowed:for any finite approximation factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ , at least 𝑛 Ω(log 𝑛 ) gates are necessary to approximatethe assignment problem within the factor 𝑟 .By combining the boolean bound (Result 1) with counting arguments, we show that the greedyalgorithm can beat approximating pure (min , +) DP algorithms on some minimization problems.
Result (min , +) circuits; Theorem 4.6) . There are doubly-exponentially many in 𝑛 minimization problems 𝑓 𝑛 such that the greedy algorithm solves 𝑓 𝑛 exactly, but any (min , +) circuitapproximating 𝑓 𝑛 within any finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ must have Ω( 𝑛 ) gates.Our proof of Result 1 is fairly simple, but it only gives us an “absolute” lower bound on the number ofgates, below which no tropical (min , +) circuit can approximate a given minimization problem within any factor. More interesting (and less simple), however, is the fact that, after an appropriate definition of the “semantic degree” of monotone boolean circuits (Section 7.1), also a converse of Result 1 holds:the approximation power of tropical (min , +) circuits is captured (not only lower bounded) by thecomputational power of monotone boolean circuits of bounded semantic degree. Result . A minimization problem 𝑓 can be approxi-mated within a factor 𝑟 by a tropical (min , +) circuit of size 𝑡 if and only if the boolean version of 𝑓 can be computed by a monotone boolean (∨ , ∧) circuit of size 𝑡 and semantic degree at most 𝑟 .We prove this result in Section 7 using convexity arguments. Yet another consequence of thesearguments is (see Remark 7) that, in order to show that the minimization problem on a family F ⊆ [ 𝑛 ] of feasible solutions can be 𝑟 -approximated by a (min , +) circuit of size 𝑡 , it is enough to designa monotone arithmetic (+ , ×) circuit of size ⩽ 𝑡 such that the polynomial computed by this circuithas the following two properties (where we, as customary, only consider monomials with nonzerocoefficients):(1) for every monomial ∏ 𝑖 ∈ 𝑇 𝑥 𝑑 𝑖 𝑖 there is a set 𝑆 ∈ F with 𝑆 ⊆ 𝑇 ;(2) for every set 𝑆 ∈ F there is a monomial ∏ 𝑖 ∈ 𝑇 𝑥 𝑑 𝑖 𝑖 with 𝑇 = 𝑆 and all 𝑑 𝑖 ⩽ 𝑟 .That is, we can approximate minimization problems by designing monotone arithmetic circuits ofbounded degree. This is a (rough) upper bound on the size of approximating (min , +) circuits in termsof arithmetic circuits. Result 3 gives a tight bound, but in terms of boolean circuits. Maximization.
It turned out that not only the approximation behaviors of (min , +) and (max , +) cir-cuits are different (approximation factors may be unbounded in the former model, while they are alwaysbounded in the latter model), but also the task of proving lower bounds for approximating (max , +) circuits is by far more difficult than that for (min , +) circuits.The point is that for approximating (max , +) circuits, even Shannon type counting arguments fail(see Section 5.1). In particular, there are doubly-exponentially many in 𝑛 maximization problems 𝑓 𝑛 such that (max , +) circuits require Ω( 𝑛 ) gates to solve any of them exactly (within the factor 𝑟 = 1 ),but one single (max , +) circuit of size 𝑂 ( 𝑛 ) approximates each of these problems within a just slightlylarger than factor 𝑟 = 1 + 𝑜 (1) (Proposition 5.2). Such a jump in circuit size occurs also on random maximization problems (Proposition 5.3). Moreover, there are also explicit maximization problems 𝑓 𝑛 that require (max , +) circuits of size at least 𝑛 ∕4 to solve them exactly (within factor 𝑟 = 1 ), butcan be approximated within the factor 𝑟 = 2 by using only 𝑛 gates (Theorem 5.4).Being warned by these facts, we go much deeper (than in the case of minimization) into the structureof approximating (max , +) circuits and prove a general “rectangle lower bound” for them.Let F be a family of feasible solutions. A rectangle is a family of sets specified by a pair A , B offamilies satisfying 𝐴 ∩ 𝐵 = ∅ for all 𝐴 ∈ A and 𝐵 ∈ B . The rectangle R = A ∨ B itself consists of allsets 𝐴 ∪ 𝐵 with 𝐴 ∈ A and 𝐵 ∈ B . The rectangle R lies below F if every set of R is contained in atleast one set of F . Given an approximation factor 𝑟 ⩾ , we say that a set 𝐹 ∈ F appears 𝑟 -balanced in the rectangle R if there are sets 𝐴 ∈ A and 𝐵 ∈ B such that 𝐹 shares ⩾ | 𝐹 | ∕ 𝑟 elements with 𝐴 ∪ 𝐵 ,and ⩾ | 𝐹 | ∕3 𝑟 elements with both 𝐴 and 𝐵 . Result . If in any rectangle lying below F , at most a 𝑡 portion of sets of F appear 𝑟 -balanced, then every (max , +) circuit approximating the maximizationproblem on F within the factor 𝑟 must have at least 𝑡 gates.Using the rectangle bound, we show that already a slight decrease of the allowed approximationfactors 𝑟 can make tractable problems intractable, and that this happens for arbitrarily large factors 𝑟 .In the following result formalizing this phenomenon, 𝜖 > is an arbitrarily small constant. Result . For every prime power 𝑚 and integer ⩽ 𝑑 ⩽ 𝑚 ,there is an explicit maximization problem 𝑓 𝑛 on 𝑛 = 𝑚 ground elements which can be approximatedwithin the factor 𝑟 = 𝑚 ∕ 𝑑 by a (max , +) circuit of size 𝑛 , but any (max , +) circuit approximating 𝑓 𝑛 within the factor (1 − 𝜖 ) 𝑟 must have at least 𝑛 𝜖𝑑 ∕4 gates.Finally, using the rectangle bound, we show that there are explicit maximization problems 𝑓 𝑛 suchthat (max , +) circuits of polynomial in 𝑛 size cannot achieve even an exponentially larger factor thanthe factor achieved by the greedy algorithm on 𝑓 𝑛 . Result (max , +) circuits; Theorem 5.13) . For every integer 𝑟 ⩾ , there are explicitmaximization problems 𝑓 𝑛 such that the greedy algorithm approximates 𝑓 𝑛 within the factor 𝑟 , butevery (max , +) circuit approximating 𝑓 𝑛 within the factor 𝑟 ∕9 must have 𝑛 Ω(1) gates.Families of feasible solutions of the maximization problems 𝑓 𝑛 in Result 5 are particular combina-torial designs, while those in Result 6 are families of perfect matchings in 𝑟 -partite 𝑟 -uniform hyper-graphs. The algorithmic message.
As we already mentioned above, it was long known that for some combi-natorial optimization problems, greedy algorithms can have much worse approximation behavior thanpure DP algorithms. Thus, Results 2 and 6 imply that the approximation powers of greedy and pureDP algorithms are incomparable : on some optimization problems, pure DP algorithms can also havemuch worse approximation behavior than greedy.
Why “only” pure DP?.
In this paper, we only consider pure (min , +) and (max , +) DP algorithms.Non-pure DP algorithms may use other arithmetic operations, rounding, as well as very powerfuloperations like conditional branchings (via if-then-else constraints), argmin, argmax, etc. The presenceof such operations makes the corresponding circuit models no longer amenable for analysis usingknown mathematical tools. In particular, such DP algorithms have the full power of arithmetic circuitsas well as of unrestricted boolean (∨ , ∧ , ¬) circuits (for example, ¬ 𝑥 is a simple conditional branchingoperation if 𝑥 = 0 then else ). Let us stress that our goal is to prove (unconditional) lower bounds .In the context of this task, even proving lower bounds for exactly solving (min , + , −) circuits (tropicalcircuits with subtraction operation allowed), remains a challenge (see Section 8.4). Organization.
In Section 3, we recall the concept of sets produced by circuits, and show that whenapproximating combinatorial optimization problems, we can safely assume that tropical circuits are constant-free , that is, contain no constants as inputs (Lemma 3.2). Sections 4 to 7 are devoted to theproofs of our main results. Results 1 and 2 are proved in Section 4, and Results 4 to 6 are provedin Section 5. In Section 6, we use convexity arguments (Farkas’ lemma) to give a tight structuralconnection between the sets of feasible solutions of optimization problems to be approximated andthe sets of feasible solutions produced by approximating tropical circuits. In Section 7, we prove the converse of our boolean lower bound for approximating (min , +) circuits (Result 3). The concludingsection (Section 8) contains some open problems. In Appendix A, we recall greedy algorithms. InAppendix B, we exhibit an exponential (almost maximal possible) decrease in the size of (max , +) circuits on explicit maximization problems when going from the approximation factor 𝑟 = 1 (exactsolution) to factor 𝑟 = 2 . Notation.
Through the paper, ℕ = {0 , , , …} will denote the set of all nonnegative integers, [ 𝑛 ] ={1 , , … , 𝑛 } the set of the first 𝑛 positive integers, ℝ + the set of all nonnegative real numbers, and 𝐸 the family of all subsets of a set 𝐸 . Also, ⃗ will denote the all- vector, ⃗𝑒 𝑖 will denote the - vectorwith exactly one in the 𝑖 th position. For sets 𝐴, 𝐵 ⊆ ℝ 𝑛 of vectors, their Minkowski sum (or sumset ) is the set of vectors 𝐴 + 𝐵 = { 𝑎 + 𝑏 ∶ 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 } ⊆ ℝ 𝑛 , where 𝑎 + 𝑏 = ( 𝑎 + 𝑏 , … , 𝑎 𝑛 + 𝑏 𝑛 ) is thecomponentwise sum of vector 𝑎 and 𝑏 . That is, we add every vector of 𝐵 to every vector of 𝐴 . For areal vector 𝑎 = ( 𝑎 , … , 𝑎 𝑛 ) and a scalar 𝜆 ∈ ℝ , 𝜆 ⋅ 𝑎 stands for the vector ( 𝜆𝑎 , … , 𝜆𝑎 𝑛 ) . If 𝐴 ⊆ ℝ 𝑛 isa set of vectors, then 𝜆 ⋅ 𝐴 stands for the set of vectors { 𝜆 ⋅ 𝑎 ∶ 𝑎 ∈ 𝐴 } . The support of vector 𝑎 is theset 𝑆 𝑎 = { 𝑖 ∶ 𝑎 𝑖 ≠ of its nonzero positions.As customary, a family F of sets is an antichain if none of its sets is a proper subset of anotherset of F . For two vectors 𝑎 ∈ ℝ 𝑛 and 𝑏 ∈ ℝ 𝑛 we write 𝑎 ⩽ 𝑏 if 𝑎 𝑖 ⩽ 𝑏 𝑖 holds for all positions 𝑖 = 1 , … , 𝑛 . A set 𝐴 of vectors is an antichain if 𝑎 ⩽ 𝑎 ′ holds for no two distinct vectors 𝑎 ≠ 𝑎 ′ ∈ 𝐴 .The characteristic vector of a set 𝑆 ⊆ [ 𝑛 ] is the vector 𝑎 ∈ {0 , 𝑛 with 𝑎 𝑖 = 1 if and only if 𝑖 ∈ 𝑆 .3. P RELIMINARIES
Every finite set
𝐴 ⊂ ℕ 𝑛 of feasible solutions defines a discrete optimization problem of the form 𝑓 ( 𝑥 ) = min 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ or of the form 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ , where here and in what follows, ⟨ 𝑎, 𝑥 ⟩ = 𝑎 𝑥 + ⋯ + 𝑎 𝑛 𝑥 𝑛 stands for the scalar product of vectors 𝑎 = ( 𝑎 , … , 𝑎 𝑛 ) and 𝑥 = ( 𝑥 , … , 𝑥 𝑛 ) .We will refer to such problems as problems defined by 𝐴 , or as problems on 𝐴 . Such a problemis a - optimization problem if the set 𝐴 ⊆ {0 , 𝑛 of feasible solutions consists of only - vectors.These latter problems are exactly what we called “combinatorial optimization” problems on families F ⊆ [ 𝑛 ] of feasible solutions, where F consists of all sets 𝑆 𝑎 = { 𝑖 ∶ 𝑎 𝑖 = 1} for vectors 𝑎 ∈ 𝐴 .To avoid trivialities, we will throughout assume that the all- vector (or the empty set) is not afeasible solution, that is, we will always assume that ⃗ 𝐴 and ∅ ∉ F .3.1. Circuits over semirings.
Recall that a (commutative) semiring is a set 𝑅 closed under two asso-ciative and commutative binary operations “addition” ( ⊕ ) and “multiplication” ( ⊗ ) , where multiplica-tion distributes over addition: 𝑥 ⊗ ( 𝑦 ⊕ 𝑧 ) = ( 𝑥 ⊗ 𝑦 ) ⊕ ( 𝑥 ⊗ 𝑧 ) . That is, in a semiring, we can “add”and “multiply” elements, but neither “subtraction” nor “division” are necessarily possible. Besides ofbeing commutative, we will assume that the semiring contains a multiplicative identity element with ⊗ 𝑥 = 𝑥 ⊗ = 𝑥 .A circuit over a semiring 𝑅 is a directed acyclic graph; parallel edges joining the same pair of nodesare allowed. Each indegree-zero node (an input node) holds either one of the variables 𝑥 , … , 𝑥 𝑛 ora semiring element. Every other node, a gate , has indegree two and performs one of the semiringoperations. One of the gates is designated as the output gate. The size of a circuit is the total numberof gates in it. A circuit is constant-free if it has no semiring elements as inputs.Since in any semiring ( 𝑅, ⊕ , ⊗ ) , multiplication distributes over addition, each circuit Φ over 𝑅 computes (at the output gate) some polynomial(1) Φ( 𝑥 , … , 𝑥 𝑛 ) = ∑ 𝑏 ∈ 𝐵 𝜆 𝑏 𝑋 𝑏 with 𝑋 𝑏 = 𝑛 ∏ 𝑖 =1 𝑥 𝑏 𝑖 𝑖 over 𝑅 in a natural way, where 𝐵 ⊂ ℕ 𝑛 is some set of exponent vectors , and 𝑥 𝑘𝑖 stands for 𝑥 𝑖 ⊗ 𝑥 𝑖 ⊗ ⋯ ⊗ 𝑥 𝑖 𝑘 -times. Since we only consider semirings with multiplicative identity, coefficients 𝜆 𝑏 ∈ 𝑅 aresemiring elements. To see why this assumption is necessary, consider the semiring ( 𝑅, + , ×) , where 𝑅 is the set of all positive even integers. Then the coefficient of the monomial 𝑥 in the polynomial 𝑥 + 𝑥 + 𝑥 is not a semiring element.In this paper, we will mainly consider circuits over three commutative and idempotent semirings ( 𝑅, ⊕ , ⊗ ) . In the boolean (∨ , ∧) semiring, we have 𝑅 = {0 , , 𝑥 ⊕ 𝑦 ∶= 𝑥 ∨ 𝑦 and 𝑥 ⊗ 𝑦 ∶= 𝑥 ∧ 𝑦 . Inthe tropical (min , +) semiring, we have 𝑅 = ℝ + , 𝑥 ⊕ 𝑦 ∶= min( 𝑥, 𝑦 ) and 𝑥 ⊗ 𝑦 ∶= 𝑥 + 𝑦 . Similarly,in the tropical (max , +) semiring, we have 𝑅 = ℝ + , 𝑥 ⊕ 𝑦 ∶= max( 𝑥, 𝑦 ) and 𝑥 ⊗ 𝑦 ∶= 𝑥 + 𝑦 . The multiplicative identity element in the boolean semiring is = 1 , and is = 0 in both tropical semirings.Over the boolean semiring, the polynomial Eq. (1) computes the monotone boolean function Φ( 𝑥 ) = ⋁ 𝑏 ∈ 𝐵 ⋀ 𝑖 ∶ 𝑏 𝑖 ≠ 𝑥 𝑖 . Over the tropical semirings, every monomial 𝑋 𝑏 = ∏ 𝑛𝑖 =1 𝑥 𝑏 𝑖 𝑖 turns into the scalar product 𝑋 𝑏 = ∑ 𝑛𝑖 =1 𝑏 𝑖 𝑥 𝑖 = ⟨ 𝑏, 𝑥 ⟩ of vectors 𝑏 and 𝑥 . Hence, the polynomial Eq. (1) solves one of the two optimizationproblems with linear objective functions:(2) Φ( 𝑥 ) = min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ + 𝜆 𝑏 or Φ( 𝑥 ) = max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ + 𝜆 𝑏 . Note that if a tropical circuit Φ is constant-free, then 𝜆 𝑏 = 0 holds for all 𝑏 ∈ 𝐵 .3.2. Sets of vectors produced by circuits.
A simple, but important in our later analysis, observationis that every circuit of 𝑛 variables over a semiring ( 𝑅, ⊕ , ⊗ ) not only computes some polynomial over 𝑅 , but also produces (purely syntactically) a finite set of vectors in ℕ 𝑛 in a natural way.At each input node holding a semiring element, the same set { ⃗ is produced. At an input nodeholding a variable 𝑥 𝑖 , the set { ⃗𝑒 𝑖 } is produced. At an “addition” ( ⊕ ) gate, the union of sets producedat its inputs is produced. Finally, at a “multiplication” ( ⊗ ) gate, the Minkowski sum of sets producedat its inputs is produced. The set produced by the entire circuit is the set produced at its output gate.It is clear that the same circuit Φ with only “addition” ( ⊕ ) and “multiplication” ( ⊗ ) gates may compute different functions over different semirings. It is, however, important to note that the set 𝐵 ⊂ ℕ 𝑛 of vectors produced by Φ is always the same—it only depends on the circuit itself, not on theunderlying semiring.On the other hand, up to coefficients, the polynomial function computed by the circuit Φ is deter-mined by the set of produced vectors. Proposition 3.1. If 𝐵 ⊂ ℕ 𝑛 is the set of vectors produced by a circuit Φ over a semiring 𝑅 , then Φ computes some polynomial over 𝑅 whose set of exponent vectors coincides with 𝐵 .Proof. Simple induction on the size of a circuit Φ . Let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced, and 𝑓 ∶ 𝑅 𝑛 → 𝑅 the polynomial function computed by Φ .If the circuit Φ consists of a single input node holding a semiring element 𝜆 ∈ 𝑅 , then 𝑓 ( 𝑥 ) = 𝜆 isa constant polynomial with a single exponent vector ⃗ . If Φ consists of a single input node holding avariable 𝑥 𝑖 , then 𝑓 ( 𝑥 ) = 𝑥 𝑖 is a degree- polynomial with the single exponent vector ⃗𝑒 𝑖 .Now, the set of exponent vectors of a sum of two polynomials is just the union of the sets of exponentvectors of these polynomials. Finally, when multiplying two polynomials, we multiply each monomialof the first polynomial with all monomials of the second polynomial. The exponent vector of a productof two monomials is the sum of exponent vectors of these monomials. (cid:3) Remark . In general, Proposition 3.1 has no converse, even for constant-free circuits: if a circuit Φ computes some polynomial 𝑓 , then Φ does not need to produce the set of exponent vectors of 𝑓 ; asimple example for tropical circuits is given in Fig. 2. Monotone arithmetic (+ , ×) circuits, that is,circuits over the arithmetic semiring ( ℝ + , + , ×) , are here an exception: for them, also the converseof Proposition 3.1 holds. Namely, if such a circuit computes a polynomial 𝑓 , then the set of vectors produced by the circuit is exactly the set of exponent vectors of this polynomial 𝑓 . This holds because,if two arithmetic polynomials coincide on sufficiently many (with respect to the number of variablesand the degrees of these polynomials) inputs, then these polynomials must syntactically coincide (evenup to coefficients). 𝑥 𝑦 + + GFED@ABC min (cid:11) (cid:19) (cid:11) (cid:19) (cid:31) (cid:31) ❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧ 𝑥 𝑦
GFED@ABC min + ! ! ❈❈❈❈ } } ④④④④ (cid:11) (cid:19) F IGURE
2. Two constant-free (min , +) circuits solving the minimization problem 𝑓 ( 𝑥, 𝑦 ) = min{2 𝑥, 𝑦 } whose set of feasible solutions is 𝐴 = {(2 , , (0 , . Thefirst circuit produces the set 𝐴 itself, whereas the second saves one gate by producinga different set 𝐵 = {(2 , , (1 , , (0 , . Here ⇓ stands for two parallel edges.3.3. Eliminating constant inputs.
Recall that an optimization problem on a set
𝐴 ⊂ ℕ 𝑛 of feasiblesolutions is of the form 𝑓 ( 𝑥 ) = min 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ or of the form 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . To avoid trivialities,we always assume that 𝐴 ≠ ∅ and ⃗ 𝐴 .These problems are “constant-free” in that they are completely specified by their sets 𝐴 of feasiblesolutions: there are no additional constant terms. In contrast, since tropical circuits can have constantinputs, the optimization problems actually solved by such circuits (exactly) may be not constant-free:they may have additional constant terms; see Eq. (2).However, as the following lemma shows, when dealing with tropical circuits approximating (constant-free) optimization problems, we can safely restrict ourselves to constant-free circuits. Recall thatconstant-free circuits only use the variables 𝑥 , … , 𝑥 𝑛 as inputs. Lemma 3.2 (Eliminating constant inputs) . If an optimization problem on a set
𝐴 ⊂ ℕ 𝑛 can be 𝑟 -approx-imated by a tropical circuit of size 𝑡 , then this problem can also be 𝑟 -approximated by a constant-freetropical circuit of size 𝑡 .Proof. Let Φ be a tropical (max , +) or (min , +) circuit, and 𝐵 ⊂ ℕ 𝑛 the set of vectors produced by Φ .By Proposition 3.1, the circuit computes the maximum or the minimum, over all vectors 𝑏 ∈ 𝐵 , oflinear functions ⟨ 𝑏, 𝑥 ⟩ + 𝜆 𝑏 , where 𝜆 𝑏 ∈ ℝ + are some constants.We obtain the constant-free version Φ ∗ of Φ as follows. First, replace every constant input by .Then eliminate zeros by repeatedly replacing gates 𝑢 +0 and max( 𝑢, by the gate 𝑢 , and a gate min( 𝑢, by an input node holding . Since Φ( 𝑥 ) ≠ must hold for at least one 𝑥 ∈ ℝ 𝑛 + , the constant input alsodisappears at the end of this replacement. Since constant inputs can only affect the additive constantterms 𝜆 𝑏 , the constant-free version Φ ∗ computes the maximum or the minimum of linear functions ⟨ 𝑏, 𝑥 ⟩ without any constant terms. Our goal is to show that Φ ∗ still 𝑟 -approximates our optimizationproblem 𝑓 on the set 𝐴 . Case Φ is a (max , +) circuit; hence, 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . In this case, we have that Φ ∗ ( 𝑥 ) =max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ , and Φ( 𝑥 ) = max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ + 𝜆 𝑏 for some nonnegative constants 𝜆 𝑏 ∈ ℝ + . Since Φ approximates 𝑓 , Φ( 𝑥 ) ⩽ 𝑓 ( 𝑥 ) must hold for all input weightings 𝑥 ∈ ℝ 𝑛 + . Taking 𝑥 = ⃗ , we obtain Φ( ⃗ ⩽ 𝑓 ( ⃗
0) = 0 and, hence, 𝜆 𝑏 = 0 for all 𝑏 ∈ 𝐵 . Thus, in the case of maximization, the constant-freeversion of the circuit solves just the same problem as the original circuit, and we are done. Case Φ is a (min , +) circuit; hence, 𝑓 ( 𝑥 ) = min 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . Since Φ 𝑟 -approximates 𝑓 , we knowthat the inequalities 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) ⩽ 𝑟 ⋅ 𝑓 ( 𝑥 ) must hold for all 𝑥 ∈ ℝ 𝑛 + . We have to show that Φ ∗ alsosatisfies these inequalities. We know that Φ ∗ ( 𝑥 ) = min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ , and Φ( 𝑥 ) = min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ + 𝜆 𝑏 forsome nonnegative constants 𝜆 𝑏 ∈ ℝ + . Since the constants 𝜆 𝑏 are nonnegative, we clearly have Φ ∗ ( 𝑥 ) ⩽ Φ( 𝑥 ) and, hence, also Φ ∗ ( 𝑥 ) ⩽ 𝑟 ⋅ 𝑓 ( 𝑥 ) for all 𝑥 ∈ ℝ 𝑛 + . So, it remains to show that Φ ∗ ( 𝑥 ) ⩾ 𝑓 ( 𝑥 ) holds for all 𝑥 ∈ ℝ 𝑛 + , as well. Weknow that Φ( 𝑥 ) ⩾ 𝑓 ( 𝑥 ) holds for all 𝑥 ∈ ℝ 𝑛 + .Assume contrariwise that Φ ∗ ( 𝑥 ) < 𝑓 ( 𝑥 ) holds for some input weighting 𝑥 ∈ ℝ 𝑛 + . Then thedifference 𝑑 = 𝑓 ( 𝑥 )−Φ ∗ ( 𝑥 ) is positive. We also know that 𝜆 ∶= max 𝑏 ∈ 𝐵 𝜆 𝑏 is positive, for otherwise,there would be nothing to prove. So, take the constant 𝑐 ∶= 2 𝜆 ∕ 𝑑 > , and consider the inputweighting 𝑧 ∶= 𝑐 ⋅ 𝑥 . Since Φ ∗ ( 𝑥 ) = 𝑓 ( 𝑥 ) − 𝑑 , and since Φ( 𝑥 ) ⩽ Φ ∗ ( 𝑥 ) + 𝜆 holds for all weightings 𝑥 ∈ ℝ 𝑛 + , the desired contradiction follows: Φ( 𝑧 ) = Φ( 𝑐 ⋅ 𝑥 ) ⩽ Φ ∗ ( 𝑐 ⋅ 𝑥 ) + 𝜆 = 𝑐 ⋅ Φ ∗ ( 𝑥 ) + 𝜆 = 𝑐 ⋅ [ 𝑓 ( 𝑥 ) − 𝑑 ] + 𝜆 = 𝑐 ⋅ 𝑓 ( 𝑥 ) − 𝑐 ⋅ 𝑑 + 𝜆 = 𝑓 ( 𝑐 ⋅ 𝑥 ) − 𝜆 = 𝑓 ( 𝑧 ) − 𝜆 < 𝑓 ( 𝑧 ) . (cid:3)
4. A
PPROXIMATION LIMITATIONS OF ( MIN ,+)
CIRCUITS
In this section, we first prove a general “boolean bound” for approximating (min , +) circuits: ifthe boolean (decision) version of a minimization problem requires monotone boolean (∨ , ∧) circuitsof size at least 𝑡 , then no (min , +) circuit of size < 𝑡 can approximate the problem within any finitefactor (Theorem 4.3). Together with known lower bounds on the monotone boolean circuit complexity,this gives us explicit minimization problems which are hard to approximate by (min , +) circuits and,hence, by pure DP algorithms; three selected examples are given in Section 4.2. Then, in Section 4.3,we combine the boolean bound (Theorem 4.3) with counting arguments to show that greedy algorithmscan “hardly” beat pure DP algorithms: there exist many minimization problems solvable by the greedyalgorithm exactly, while polynomial-size (min , +) circuits cannot approximate any of them within anyfinite factor.4.1. The boolean bound for approximating (min , +) circuits. Recall that the support of a vector 𝑎 ∈ ℕ 𝑛 is the set 𝑆 𝑎 = { 𝑖 ∶ 𝑎 𝑖 ≠ of its nonzero positions. Every finite set 𝐴 ⊂ ℕ 𝑛 of vectors defines the monotone boolean function 𝑓 𝐴 ( 𝑥 ) = ⋁ 𝑎 ∈ 𝐴 ⋀ 𝑖 ∈ 𝑆 𝑎 𝑥 𝑖 . Note that, for every input 𝑥 ∈ {0 , 𝑛 , we have(3) 𝑓 𝐴 ( 𝑥 ) = 1 if and only if 𝑆 𝑥 ⊇ 𝑆 𝑎 for some 𝑎 ∈ 𝐴 .For example, if 𝐴 is the set of characteristic - vectors of perfect matchings in 𝐾 𝑚,𝑚 , then 𝑓 acceptsa subgraph 𝐺 of 𝐾 𝑚,𝑚 if and only if 𝐺 contains a perfect matching.Two sets 𝐴, 𝐵 ⊆ ℕ 𝑛 are similar if the support of every vector 𝑏 ∈ 𝐵 contains the support of at leastone vector 𝑎 ∈ 𝐴 , and vice versa. That is, 𝐴 and 𝐵 are similar if and only if(4) ∀ 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 ∶ 𝑆 𝑏 ⊇ 𝑆 𝑎 and ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 ∶ 𝑆 𝑎 ⊇ 𝑆 𝑏 .Observation Eq. (3) immediately yields the following. Proposition 4.1.
Two sets of vectors define the same boolean function if and only if these sets aresimilar.
The main connection between approximating (min , +) circuits and monotone boolean circuits isgiven by the following lemma. The boolean version of a constant-free tropical (min , +) circuit is themonotone boolean (∨ , ∧) circuit obtained by replacing each min -gate by an ∨ -gate, and each + -gateby an ∧ -gate. Lemma 4.2.
If a constant-free (min , +) circuit Φ approximates the minimization problem on a set 𝐴 ⊂ ℕ 𝑛 within a finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ , then the boolean version of Φ computes the booleanfunction defined by 𝐴 .Proof. Let
𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by Φ . Since the circuit Φ is constant-free, it solvesthe minimization problem Φ( 𝑥 ) = min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ defined by this set 𝐵 . The minimization problem on 𝐴 is 𝑓 ( 𝑥 ) = min 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . We know that 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) ⩽ 𝑟 ⋅ 𝑓 ( 𝑥 ) must hold for all input weightings 𝑥 ∈ ℝ 𝑛 + . The boolean version 𝜙 of Φ also produces the same set 𝐵 . By Proposition 4.1, it remains toshow that the set 𝐵 is similar to 𝐴 ; see Eq. (4).For the sake of contradiction, suppose first that there is a vector 𝑏 ∈ 𝐵 such that 𝑆 𝑎 ⧵ 𝑆 𝑏 ≠ ∅ holdsfor all vectors 𝑎 ∈ 𝐴 . Consider the assignment 𝑥 ∈ {0 , 𝑛 of weights such that 𝑥 𝑖 = 0 for 𝑖 ∈ 𝑆 𝑏 , and 𝑥 𝑖 = 1 for 𝑖 ∉ 𝑆 𝑏 . On this weighting, we have Φ( 𝑥 ) ⩽ ⟨ 𝑏, 𝑥 ⟩ = 0 . But since every vector 𝑎 ∈ 𝐴 has aposition 𝑖 ∉ 𝑆 𝑏 with 𝑎 𝑖 ≠ , ⟨ 𝑎, 𝑥 ⟩ ⩾ holds for all 𝑎 ∈ 𝐴 and, hence, also 𝑓 ( 𝑥 ) ⩾ , contradictingthe inequality 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) .Now suppose that there is a vector 𝑎 ∈ 𝐴 such that 𝑆 𝑏 ⧵ 𝑆 𝑎 ≠ ∅ holds for all vectors 𝑏 ∈ 𝐵 . Let 𝑀 = max{ ⟨ 𝑎, 𝑎 ⟩ ∶ 𝑎 ∈ 𝐴 } , and consider the weighting 𝑥 ∈ {1 , 𝑟𝑀 +1} 𝑛 such that 𝑥 𝑖 = 1 for all 𝑖 ∈ 𝑆 𝑎 and 𝑥 𝑖 = 𝑟𝑀 + 1 for all 𝑖 ∉ 𝑆 𝑎 (note that 𝑟𝑀 + 1 is a finite number, because both the approximationfactor 𝑟 and the set 𝐴 are finite. Then 𝑓 ( 𝑥 ) ⩽ ⟨ 𝑎, 𝑥 ⟩ = ⟨ 𝑎, 𝑎 ⟩ ⩽ 𝑀 . But since every vector 𝑏 ∈ 𝐵 hasa position 𝑖 ∉ 𝑆 𝑎 such that 𝑏 𝑖 ⩾ , we have Φ( 𝑥 ) ⩾ 𝑟𝑀 + 1 > 𝑟 ⋅ 𝑓 ( 𝑥 ) , contradicting the inequality Φ( 𝑥 ) ⩽ 𝑟 ⋅ 𝑓 ( 𝑥 ) . (cid:3) For a set
𝐴 ⊆ ℕ 𝑛 of vectors, let 𝖡𝗈𝗈𝗅 ( 𝐴 ) denote the minimum size of a monotone boolean (∨ , ∧) circuit computing the boolean function 𝑓 𝐴 defined by 𝐴 . Let also 𝖬𝗂𝗇 𝑟 ( 𝐴 ) denote the minimum sizeof a tropical (min , +) circuit approximating the minimization problem on 𝐴 within the factor 𝑟 . Theorem 4.3 (Boolean bound) . For every finite set
𝐴 ⊂ ℕ 𝑛 and every finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ , wehave 𝖬𝗂𝗇 𝑟 ( 𝐴 ) ⩾ 𝖡𝗈𝗈𝗅 ( 𝐴 ) .Proof. Take a (min , +) circuit Φ of size 𝑡 = 𝖬𝗂𝗇 𝑟 ( 𝐴 ) approximating the minimization problem on 𝐴 within the factor 𝑟 , and let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by Φ . By Lemma 3.2, we can assumethat the circuit Φ is constant-free. Hence, by Lemma 4.2, the boolean version 𝜙 of Φ (which has thesame size) computes the boolean function defined by the set 𝐴 , as desired. (cid:3) Remark . Note that Theorem 4.3 does not exclude that, using more than
𝖡𝗈𝗈𝗅 ( 𝐴 ) gates, (min , +) circuits could achieve finite (and even small) approximation factors. The boolean bound 𝖡𝗈𝗈𝗅 ( 𝐴 ) isjust an “absolute” lower bound below which no approximation is possible at all. Remark . The proof of Theorem 4.3 is so direct and elementary, because it totally ignores the givenapproximation factor 𝑟 : it only must be finite and, hence, can be used in input weightings to fool toosmall (min , +) circuits. Using more involved arguments (based on Farkas’ lemma), we will show inSection 7 (Theorem 7.1) that, under an appropriate definition of the “semantic degree” of monotoneboolean circuits, Theorem 4.3 has also a converse : a minimization problem can be approximated withina factor 𝑟 by a tropical (min , +) circuit of size 𝑡 if and only if the boolean version of this problem canbe computed by a monotone boolean (∨ , ∧) circuit of size 𝑡 and semantic degree at most 𝑟 . Thus,the approximation power of tropical (min , +) circuits is captured , not only lower bounded, by the computational power of monotone boolean circuits.4.2. Explicit lower bounds.
Together with lower bounds on the monotone boolean circuit complexity,the boolean bound (Theorem 4.3) immediately yields the same lower bounds on the size of approxi-mating (min , +) circuits. Let us mention some examples. In the lightest triangle problem, we are given an assignment of nonnegative weights to the edges of 𝐾 𝑛 , and the goal is to compute the minimum weight of a triangle. Corollary 4.1.
The lightest triangle problem in 𝐾 𝑛 can be solved by a (min , +) circuit using only 𝑛 gates, but no (min , +) circuit with 𝑛 gates can approximate this problem within any finite factor.Proof. Since we only have ( 𝑛 ) triangles, a trivial (min , +) circuit of size at most 𝑛 (taking the mini-mum over all triangles) solves this problem exactly. On the other hand, it is known ([1, Lemma 3.14])that the decision version of this problem requires monotone boolean circuits with Ω( 𝑛 ∕ log 𝑛 ) gates.Theorem 4.3 gives the same lower bound for approximating (min , +) circuits. (cid:3) Recall that the 𝑛 - assignment problem is: given an assignment of nonnegative real weights to theedges of the complete bipartite 𝑛 × 𝑛 graph, compute the minimum weight of a perfect matching inthis graph. The corresponding family of feasible solutions is here the family of all perfect matchings,viewed as sets of their edges. Corollary 4.2.
Every (min , +) circuit approximating the 𝑛 -assignment problem within any finite factormust have at least 𝑛 Ω(log 𝑛 ) gates.Proof. The boolean function defined by the family of feasible solutions of the assignment problemis the boolean permanent function which, as proved by Razborov [26], requires monotone booleancircuits of size 𝑛 Ω(log 𝑛 ) . (cid:3) Let 𝑛 be a prime power, and ⩽ 𝑑 ⩽ 𝑛 an integer. The polynomial ( 𝑛, 𝑑 ) -design is the family of all | F | = 𝑛 𝑑 𝑛 -element subsets {( 𝑎, 𝑝 ( 𝑎 )) ∶ 𝑎 ∈ GF( 𝑛 )} of the grid GF( 𝑛 ) × GF( 𝑛 ) , where 𝑝 = 𝑝 ( 𝑥 ) rangesover all 𝑛 𝑑 univariate polynomials of degree at most 𝑑 − 1 over GF( 𝑛 ) . Corollary 4.3. If 𝑑 ⩽ ( 𝑛 ∕4 ln 𝑛 ) , then every (min , +) circuit approximating the minimization prob-lem on the polynomial ( 𝑛, 𝑑 ) -design within any finite factor must have at least 𝑛 Ω( 𝑑 ) gates.Proof. By (numerically) improving the earlier result of Andreev [2], Alon and Boppana [1] have shownthat, at least for such values of 𝑑 , any monotone boolean circuit computing the boolean function definedby the corresponding family of feasible solutions requires 𝑛 Ω( 𝑑 ) gates. (cid:3) Greedy can beat approximating (min,+) circuits.
Our goal now is to show that there exist many(combinatorial) minimization problems which are solvable by the greedy algorithm exactly (within thefactor 𝑟 = 1 ), but no (min , +) circuit with a polynomial in the number 𝑛 of ground elements numberof gates can approximate any of these problems within any finite factor 𝑟 = 𝑟 ( 𝑛 ) .We identify matroids with their families of bases. Under this proviso, a family F is a matroid ifand only if F is uniform (all sets have the same cardinality) and the basis exchange axiom holds: if 𝐴 ≠ 𝐵 ∈ F , then for every 𝑎 ∈ 𝐴 ⧵ 𝐵 there is a 𝑏 ∈ 𝐵 ⧵ 𝐴 such that the set ( 𝐴 ⧵ { 𝑎 }) ∪ { 𝑏 } belongs to F . It is well known (see, for example, [23, Theorem 1.8.4]) that an optimization problem on an antichain F can be solved by the greedy algorithm exactly if and only if F is a matroid. This fact is usually calledthe Rado–Edmonds theorem [25, 7]. In contrast, we will now show that most matroids require (min , +) circuits of exponential size to be even only approximated within any finite factor. We will do this bycounting, so we need a lower bound on the number of matroids.The following simple construction of matroids was implicit in several papers, starting from those ofPiff and Welsh [24], and Knuth [18], and was made explicit by Bansal, Pendavingh and Van der Pol [3,Lemma 8]. Let ( [ 𝑛 ] 𝑚 ) denote the family of all 𝑚 -element subsets of [ 𝑛 ] = {1 , … , 𝑛 } . The Hammingdistance between two sets 𝐴 and 𝐵 is dist( 𝐴, 𝐵 ) = | 𝐴 ⧵ 𝐵 | + | 𝐵 ⧵ 𝐴 | . A family H is separated if dist( 𝐴, 𝐵 ) > holds for all 𝐴 ≠ 𝐵 ∈ H . Proposition 4.4. If H ⊆ ( [ 𝑛 ] 𝑚 ) is separated, then F = ( [ 𝑛 ] 𝑚 ) ⧵ H is a matroid.Proof. Suppose contrariwise that F is not a matroid. Since the family F is uniform, there must be twosets 𝐴 ≠ 𝐵 ∈ F violating the basis exchange axiom: there is an 𝑎 ∈ 𝐴 ⧵ 𝐵 such that ( 𝐴 ⧵ { 𝑎 })∪{ 𝑏 } ∉ F for all 𝑏 ∈ 𝐵 . Observe that 𝐵 ⧵ 𝐴 must have at least two elements: held 𝐵 ⧵ 𝐴 = { 𝑏 } then, since both 𝐴 and 𝐵 have the same cardinality, the set ( 𝐴 ⧵ { 𝑎 }) ∪ { 𝑏 } would coincide with 𝐵 and, hence, wouldbelong to F . So, take 𝑏 ≠ 𝑐 ∈ 𝐵 ⧵ 𝐴 and consider the sets 𝑆 = ( 𝐴 ⧵ { 𝑎 }) ∪ { 𝑏 } and 𝑇 = ( 𝐴 ⧵ { 𝑎 }) ∪ { 𝑐 } .Since the basis exchange axiom fails for 𝐴 and 𝐵 , neither 𝑆 nor 𝑇 can belong to F ; hence, both sets 𝑆 and 𝑇 belong to the family ( [ 𝑛 ] 𝑚 ) ⧵ F = H . But dist( 𝑆, 𝑇 ) = | { 𝑏, 𝑐 } | = 2 , a contradiction with thefamily H being separated. (cid:3) Proposition 4.5.
There are 𝑛𝑚 ) ∕ 𝑛 matroids F ⊆ ( [ 𝑛 ] 𝑚 ) such that ( [ 𝑛 ] 𝑚 ) ⧵ F is separated.Proof. Since subfamilies of separated families are also separated, it is enough, by Proposition 4.4 toshow that a separated family H ⊆ ( [ 𝑛 ] 𝑚 ) of size | H | ⩾ ( 𝑛𝑚 ) ∕ 𝑛 exists.The following amazingly simple argument was suggested by Graham and Sloane [13]. For 𝑙 ∈{0 , , … , 𝑛 − 1} , let H 𝑙 be the family of all sets 𝑆 ∈ ( [ 𝑛 ] 𝑚 ) such that ∑ 𝑖 ∈ 𝑆 𝑖 = 𝑙 mod 𝑛 . We claim thateach such family H 𝑙 is separated. Suppose contrariwise that dist( 𝑆, 𝑇 ) = 2 holds for some two sets 𝑆 ≠ 𝑇 of H 𝑙 . Then 𝑆 = 𝐴 ∪ { 𝑠 } and 𝑇 = 𝐴 ∪ { 𝑡 } for some ( 𝑚 − 1) -element set 𝐴 , and 𝑠 ≠ 𝑡 aredistinct numbers in [ 𝑛 ] ⧵ 𝐴 . But then for 𝑎 = ∑ 𝑖 ∈ 𝐴 𝑖 , we have 𝑎 + 𝑠 = 𝑙 mod 𝑛 and 𝑎 + 𝑡 = 𝑙 mod 𝑛 ,which is impossible because both numbers 𝑠 and 𝑡 are at most 𝑛 . Thus, every family H 𝑙 is separated.Since there are only 𝑛 such families, and they exhaust the entire family ( [ 𝑛 ] 𝑚 ) , there must be an 𝑙 forwhich | H 𝑙 | ⩾ ( 𝑛𝑚 ) ∕ 𝑛 holds. (cid:3) Theorem 4.6.
There are at least 𝑛 ∕ 𝑛 matroids F ⊆ [ 𝑛 ] such that every (min , +) circuit approximat-ing the minimization problem on any of them within any finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ must have at least 𝑛 ∕ 𝑛 gates.Proof. The number of monotone boolean (∨ , ∧) circuits of size 𝑡 on 𝑛 input variables is at most 𝐿 ( 𝑛, 𝑡 ) =2 𝑡 ( 𝑡 + 𝑛 ) 𝑡 . This is, actually, an upper bound on the number of constant-free circuits over any semiring ( 𝑅, ⊕ , ⊗ ) . Indeed, each gate in such a circuit is assigned a semiring operation (two choices) and actson two previous nodes. Each previous node can either be a previous gate (at most 𝑡 choices) or aninput variable ( 𝑛 choices). Thus, each single gate has at most 𝑁 = 2( 𝑡 + 𝑛 ) choices, and the numberof choices for a circuit is at most 𝑁 𝑡 .When applied with 𝑚 = ⌊ 𝑛 ∕2 ⌋ , Proposition 4.5 gives us at least 𝑀 ( 𝑛 ) = 2( 𝑛𝑚 ) ∕ 𝑛 ⩾ 𝑛 ∕2 𝑛 matroids F ⊆ ( [ 𝑛 ] 𝑚 ) . On the other hand, at most 𝐿 ( 𝑛, 𝑡 ) families F ⊆ [ 𝑛 ] can have monotone boolean circuitcomplexity at most 𝑡 . For 𝑡 ∶= 2 𝑛 ∕ 𝑛 , we have log 𝐿 ( 𝑛, 𝑡 ) = 2 𝑛 ∕ 𝑛 + (2 𝑛 +1 ∕ 𝑛 ) log( 𝑛 + 2 𝑛 ∕ 𝑛 ) = 𝑂 (2 𝑛 ∕ 𝑛 ) ≪ log 𝑀 ( 𝑛 ) = 2 𝑛 ∕2 𝑛 . Since every circuit computes only one function, at least 𝑀 ( 𝑛 ) − 𝐿 ( 𝑛, 𝑡 ) ⩾ 𝐿 ( 𝑛, 𝑡 ) matroids require monotone boolean circuits of size at least 𝑡 = 2 𝑛 ∕ 𝑛 . Theorem 4.3yields the same lower bound for approximating (min , +) circuits. (cid:3)
5. A
PPROXIMATION LIMITATIONS OF ( MAX ,+)
CIRCUITS
Given a family F ⊆ [ 𝑛 ] of feasible solutions, and an approximation factor 𝑟 ⩾ , we will denote by 𝖬𝖺𝗑 𝑟 ( F ) the minimum number of gates in a (max , +) circuit approximating the maximization problem 𝑓 ( 𝑥 ) = max 𝑆 ∈ F ∑ 𝑖 ∈ 𝑆 𝑥 𝑖 on F within the factor 𝑟 .In Section 4, we have shown that there are (even explicit) families F ⊆ [ 𝑛 ] , the minimization problems on which cannot be approximated by small (polynomial in 𝑛 ) size (min , +) circuits within any finite factor 𝑟 = 𝑟 ( 𝑛 ) . On the other hand, in the case of maximization problems, the approximationfactor is always finite. Namely, we always have 𝖬𝖺𝗑 𝑛 ( F ) ⩽ 𝑛 − 1 : since the weights are nonnegative,we can just use the trivial (max , +) circuit max{ 𝑥 , … , 𝑥 𝑛 } .5.1. Counting fails for approximating (max , +) circuits. There is an even more substantial differ-ence between approximating (min , +) and (max , +) circuits than just the “bounded versus unboundedapproximation factors” phenomenon: unlike for (min , +) circuits, even counting arguments are un-likely to yield large lower bounds on the size of approximating (max , +) circuits, even for very smallapproximation factors 𝑟 = 1 + 𝑜 (1) .Say that a family F ⊆ [ 𝑛 ] is 𝑘 - dense if every 𝑘 -element subset of [ 𝑛 ] is contained in at least one setof F . The top 𝑘 -of- 𝑛 selection problem 𝑓 𝑛,𝑘 ( 𝑥 , … , 𝑥 𝑛 ) outputs the sum of the 𝑘 largest input numbers. Proposition 5.1.
The top 𝑘 -of- 𝑛 selection problem 𝑓 𝑛,𝑘 can be solved by a (max , +) circuit of size 𝑘𝑛 ,and this circuit approximates the maximization problem on every 𝑘 -dense family F ⊆ ( [ 𝑛 ] 𝑚 ) within thefactor 𝑟 = 𝑚 ∕ 𝑘 .Proof. The family of feasible solutions of 𝑓 𝑛,𝑘 consists of all 𝑘 -element subsets of [ 𝑛 ] = {1 , … , 𝑛 } .In particular, 𝑓 𝑛, ( 𝑥 ) = max{ 𝑥 , … , 𝑥 𝑛 } and 𝑓 𝑛,𝑛 ( 𝑥 ) = 𝑥 + ⋯ + 𝑥 𝑛 . The Pascal identity ( 𝑛 +1 𝑘 ) = ( 𝑛𝑘 ) + ( 𝑛𝑘 −1 ) for binomial coefficients gives us the recursion 𝑓 𝑛 +1 ,𝑘 ( 𝑥 , … , 𝑥 𝑛 +1 ) = max{ 𝑓 𝑛,𝑘 ( 𝑥 , … , 𝑥 𝑛 ) , 𝑓 𝑛,𝑘 −1 ( 𝑥 , … , 𝑥 𝑛 ) + 𝑥 𝑛 +1 } . So, 𝑓 𝑛,𝑘 can be solved by a (max , +) circuit with only 𝑘𝑛 (max , +) gates.Now let F ⊆ ( [ 𝑛 ] 𝑚 ) be a 𝑘 -dense family. The maximization problem on F is 𝑓 ( 𝑥 ) = max 𝑆 ∈ F ∑ 𝑖 ∈ 𝑆 𝑥 𝑖 .Since the weights are nonnegative, the 𝑘 -denseness of F ensures that 𝑓 ( 𝑥 ) ⩾ 𝑓 𝑛,𝑘 ( 𝑥 ) . On the otherhand, since no solution has more than 𝑚 elements, the optimal weight 𝑓 ( 𝑥 ) of a feasible solutioncannot exceed 𝑚 ∕ 𝑘 times the sum of weights of 𝑘 heaviest elements in this solution. Hence, 𝑓 ( 𝑥 ) ⩽ ( 𝑚 ∕ 𝑘 ) ⋅ 𝑓 𝑛,𝑘 ( 𝑥 ) , as desired. (cid:3) Proposition 5.2.
There exist doubly-exponentially many in 𝑛 families F ⊆ [ 𝑛 ] such that 𝖬𝖺𝗑 ( F ) =2 Ω( 𝑛 ) but 𝖬𝖺𝗑 𝑜 (1) ( F ) ⩽ 𝑛 . The families F are here matroids , and the upper bound is achieved by one single (max , +) circuit. Proof.
Let 𝑛 be a sufficiently large even integer, and 𝑚 = 𝑛 ∕2 . Proposition 4.5 gives us at least 𝑀 ∶= 2( 𝑛𝑚 ) ∕ 𝑛 families F ⊆ ( [ 𝑛 ] 𝑚 ) (which are matroids) with the property that the Hamming distancebetween any two distinct sets 𝐴 ≠ 𝐵 ∈ ( [ 𝑛 ] 𝑚 ) ⧵ F is > . We claim that each such family F is 𝑘 -densefor 𝑘 ∶= 𝑚 − 1 . To see this, take any set 𝑇 ∈ ( [ 𝑛 ] 𝑚 −1 ) , any two distinct elements 𝑎 ≠ 𝑏 outside 𝑇 , andconsider the 𝑚 -element sets 𝐴 = 𝑇 ∪ { 𝑎 } and 𝐵 = 𝑇 ∪ { 𝑏 } . Since the Hamming distance between 𝐴 and 𝐵 is , they cannot both lie outside the family F . So, at least one of them must belong to F , asdesired.We thus have at least 𝑀 families F ⊆ ( [ 𝑛 ] 𝑚 ) which are 𝑘 -dense for 𝑘 = 𝑚 −1 . By Proposition 5.1, one (max , +) circuit of size at most 𝑘𝑛 ⩽ 𝑛 for the top 𝑘 -of- 𝑛 problem 𝑓 𝑛,𝑘 approximates the maximizationproblem on each of these 𝑀 families within the factor 𝑟 = 𝑚 ∕( 𝑚 − 1) = 1 + 1∕( 𝑚 − 2) = 1 + 𝑜 (1) .On the other hand, by Lemma 3.2, we can consider only constant-free (max , +) circuits, and the samecounting argument as in the proof of Theorem 4.6 yields the lower bound 𝖬𝖺𝗑 ( F ) = 2 Ω( 𝑛 ) for doubly-exponentially many of these families F . (cid:3) Remark (max , +) ) . The standard counting (as in the proofof Theorem 4.6) shows that the boolean function defined by some of the families F given by Proposi-tion 5.2 (actually, by many of these families) requires monotone boolean circuits of size Ω( 𝑛 ) , but (byProposition 5.2) 𝖬𝖺𝗑 𝑟 ( F ) ⩽ 𝑛 holds already for a factor 𝑟 = 1 + 𝑜 (1) .Actually, small (max , +) circuits can approximate even random maximization problems quite well.For an even integer 𝑛 ⩾ and 𝑚 = 𝑛 ∕2 , let F be a random family of 𝑚 -element subsets of [ 𝑛 ] witheach 𝑚 -element subset being included in F independently with probability . Proposition 5.3.
With probability 𝑜 (1) , 𝖬𝖺𝗑 𝑜 (1) ( F ) ⩽ 𝑛 holds.Proof. Let 𝑘 ∶= 𝑚 − 2 . Since each 𝑘 -element set is contained in 𝑙 = ( 𝑛 − 𝑘 ) = Ω( 𝑛 ) sets of ( [ 𝑛 ] 𝑚 ) , theprobability that a fixed 𝑘 -element set will be contained in none of the sets of F is (1∕2) 𝑙 = 2 −Ω( 𝑛 ) .So, by the union bound, the family F is not 𝑘 -dense with probability at most ( 𝑛𝑚 ) ⋅ −Ω( 𝑛 ) = 2 −Ω( 𝑛 ) .That is, the family F is 𝑘 -dense with probability at least −Ω( 𝑛 ) . By Proposition 5.1, with thisprobability, the (max , +) circuit for the top 𝑘 -of- 𝑛 selection problem (see Proposition 5.1) approximatesthe maximization problem on a random family F ⊆ ( [ 𝑛 ] 𝑛 ∕2 ) within the factor 𝑟 = 𝑚 ∕ 𝑘 = 1 + 𝑜 (1) . (cid:3) Propositions 5.2 and 5.3 only show the mere existence of maximization problem that are hard tosolve by (max , +) circuits exactly (with factor 𝑟 = 1 ), but can be approximated by small (max , +) circuits within a slightly larger factor 𝑟 > . Still, there are also explicit maximization problemsexhibiting a similar gap.A family F ⊆ [ 𝑛 ] is a Sidon family if the set
𝐴 ⊆ {0 , 𝑛 of the characteristic - vectors of sets in F has the following property for all vectors 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐴 : if 𝑎 + 𝑏 = 𝑐 + 𝑑 , then { 𝑎, 𝑏 } = { 𝑐, 𝑑 } (theaddition is over the reals, not over GF(2) ). That is, knowing the sum 𝑎 + 𝑏 ∈ {0 , , 𝑛 of two vectors 𝑎, 𝑏 ∈ 𝐴 , we know which vectors were added. Theorem 5.4 (Explicit gaps) . Let 𝑚 be an odd integer, and 𝑛 = 4 𝑚 . Then there is an explicit Sidonfamily F ⊆ [ 𝑛 ] such that 𝖬𝖺𝗑 ( F ) ⩾ 𝑛 ∕4 but 𝖬𝖺𝗑 ( F ) ⩽ 𝑛 . The lower bound here follows from known lower bounds on the monotone arithmetic circuit com-plexity of polynomials whose sets of exponent vectors are Sidon sets, but the proof of the upper boundis somewhat technical. So, since we are mainly interested in proving lower bounds, we postpone theentire proof of Theorem 5.4 to Appendix B.The message of Propositions 5.2 and 5.3 is: while most problems are hard to solve exactly, they aretrivially approximable by just one small (max , +) circuit within a small factor 𝑟 = 1 + 𝑜 (1) . Togetherwith Theorem 5.4 (and Remark 4), this serves as a serious indication that the task of proving lowerbounds on the size of approximating (max , +) circuits is by far more difficult than for (min , +) circuits:monotone boolean circuits cannot help then, and even counting arguments are unlikely to work against (max , +) circuits.Still, by looking more carefully into the structure of vectors produced by approximating (max , +) circuits (Lemma 5.5), and using structural restrictions of such sets given by a “decomposition lemma”(Lemma 5.6), we will be able to derive a general “rectangle bound” for approximating (max , +) circuits(Theorem 5.9).5.2. Structure of approximating (max , +) circuits. Since we are interested in the structure of sets of vectors produced by (approximating) circuits, it will be convenient to turn to the language of vectors . Lemma 5.5. If Φ is a (max , +) circuit approximating the maximization problem on a set 𝐴 ⊆ {0 , 𝑛 within a factor 𝑟 , then the set 𝐵 ⊂ ℕ 𝑛 of vectors produced by Φ has the following two properties: (i) if 𝑏 ∈ 𝐵 , then 𝑏 ⩽ 𝑎 for some 𝑎 ∈ 𝐴 ; (ii) if 𝑎 ∈ 𝐴 , then ⟨ 𝑎, 𝑏 ⟩ ⩾ 𝑟 ⟨ 𝑎, 𝑎 ⟩ for some 𝑏 ∈ 𝐵 .Proof. By Lemma 3.2, we can assume that the circuit Φ is constant-free. By Proposition 3.1, thecircuit Φ solves the maximization problem Φ( 𝑥 ) = max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ . The maximization problem on 𝐴 isof the form 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . Since the circuit 𝑟 -approximates the maximization problem on 𝐴 ,we know that 𝑟 ⋅ 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) ⩽ 𝑓 ( 𝑥 ) must hold for all input weightings 𝑥 ∈ ℝ 𝑛 + .Had some vector 𝑏 ∈ 𝐵 a position 𝑖 with 𝑏 𝑖 > , then on the input 𝑥 = ⃗𝑒 𝑖 , we would have Φ( 𝑥 ) ⩾ ⟨ 𝑏, 𝑥 ⟩ = 𝑏 𝑖 > but 𝑓 ( 𝑥 ) ⩽ , since all vectors in 𝐴 are - vectors. So, 𝐵 ⊆ {0 , 𝑛 , that is, the set 𝐵 also consists of only - vectors.To show item (i), suppose contrariwise that there is a vector 𝑏 ∈ 𝐵 such that 𝑏 ≰ 𝑎 holds for allvectors 𝑎 ∈ 𝐴 . Since (as we have just shown) 𝑏 is a - vector, this means that, for every vector 𝑎 ∈ 𝐴 , there is a position 𝑖 where 𝑏 𝑖 = 1 but 𝑎 𝑖 = 0 . Hence, on the weighting 𝑥 ∶= 𝑏 , we have Φ( 𝑥 ) ⩾ ⟨ 𝑏, 𝑥 ⟩ = ⟨ 𝑏, 𝑏 ⟩ , but ⟨ 𝑎, 𝑥 ⟩ = ⟨ 𝑎, 𝑏 ⟩ ⩽ ⟨ 𝑏, 𝑏 ⟩ − 1 for all 𝑎 ∈ 𝐴 , a contradiction with Φ( 𝑥 ) ⩽ 𝑓 ( 𝑥 ) .To show item (ii), assume contrariwise that there is some vector 𝑎 ∈ 𝐴 such that ⟨ 𝑎, 𝑏 ⟩ < 𝑚 ∕ 𝑟 holds for all vectors 𝑏 ∈ 𝐵 , where 𝑚 = ⟨ 𝑎, 𝑎 ⟩ . Then, on the input 𝑥 ∶= 𝑎 , we have Φ( 𝑥 ) < 𝑚 ∕ 𝑟 but 𝑓 ( 𝑥 ) ⩾ ⟨ 𝑎, 𝑎 ⟩ = 𝑚 , a contradiction with 𝑟 ⋅ 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) . (cid:3) Remark . Note that Lemma 5.5 holds even when the circuit Φ is only required to 𝑟 -approximatethe given minimization problem on input weightings 𝑥 ∈ {0 , 𝑛 . Indeed, to eliminate constant in-puts from (max , +) circuits in Lemma 3.2 we only used the input weighting 𝑥 = ⃗ , and the proof ofLemma 5.5 itself also uses only boolean - weightings. This implies that the rectangle bound (Theo-rem 5.9), whose proof will use Lemma 5.5, holds also when the (max , +) circuits must approximate agiven problem only on boolean - weightings.5.3. Minkowski circuits.
As we already mentioned in Section 3.2, unlike the function computed bya circuit Φ over a semiring ( 𝑅, ⊕ , ⊗ ) , the set 𝐵 ⊂ ℕ 𝑛 of vectors produced by Φ does not dependon the underlying semiring—it only depends on the circuit itself. That is, 𝐵 depends only on whatthe underlying graph of Φ is, and what of the two semiring operations are associated with gates. Thisindependence of produced sets from actual semirings is captured by the model of “Minkowski circuits.”These circuits allow one to analyze the structure of sets produced by circuits over arbitrary semiringsin a uniform and mathematically clean way.A Minkowski circuit Φ is a directed acyclic graph with 𝑛 + 1 input (indegree zero) nodes holdingsingle-element sets { ⃗ , { ⃗𝑒 } , … , { ⃗𝑒 𝑛 } . Every other node, a gate , has indegree two, and performseither the set-theoretic union ( ∪ ) or the Minkowski sum ( + ) operation on its two inputs.The sets 𝑋 𝑣 ⊂ ℕ 𝑛 of vectors produced at the gates 𝑣 of Φ are obtained as follows. If 𝑣 is an inputnode, then 𝑋 𝑣 is one of the single-element sets { ⃗ , { ⃗𝑒 } , … , { ⃗𝑒 𝑛 } , depending on which of these sets isheld by the node 𝑣 . Then 𝑋 𝑣 = 𝑋 𝑢 ∪ 𝑋 𝑤 if 𝑣 = 𝑢 ∪ 𝑤 is a union gate, and 𝑋 𝑣 = 𝑋 𝑢 + 𝑋 𝑤 if 𝑣 = 𝑢 + 𝑤 is a Minkowski sum gate. The set 𝐵 ⊂ ℕ 𝑛 produced by the entire circuit Φ is the set 𝑋 𝑣 produced atthe output gate 𝑣 .The Minkowski version of a circuit Φ over an arbitrary semiring ( 𝑅, ⊕ , ⊗ ) is obtained by replacingeach input constant 𝜆 ∈ 𝑅 by the singleton { ⃗ , each input variable 𝑥 𝑖 by the singleton { ⃗𝑒 𝑖 } , each“addition” ( ⊕ ) gate by the union (∪) gate, and each “multiplication” ( ⊗ ) gate by the Minkowski sum (+) gate.The model of Minkowski circuits is justified by the following trivial observation, which followsdirectly from the definition of sets produced by circuits over semirings: the set produced by a circuitover any semiring is the set produced by the Minkowski version of this circuit. Decomposition lemma for Minkowski circuits.
We will prove lower bounds for approximating (max , +) circuits using a general “decomposition lemma” for Minkowski circuits. The sumset definedby two sets 𝑋 ⊆ ℕ 𝑛 and 𝑌 ⊆ ℕ 𝑛 of vectors is the Minkowski sum 𝑋 + 𝑌 = { 𝑥 + 𝑦 ∶ 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 } of these two sets.Sumsets naturally emerge in every Minkowski circuit Φ . At each Minkowski sum gate following agate 𝑣 (if there is any), the set 𝑋 𝑣 of vectors produced at 𝑣 is “enlarged” by adding at least one vector to all vectors in 𝑋 𝑣 . So, when we arrive at the output gate 𝑤 , the entire translates 𝑋 𝑣 + 𝑦 = { 𝑥 + 𝑦 ∶ 𝑥 ∈ 𝑋 𝑣 } of 𝑋 𝑣 by some vectors 𝑦 ∈ ℕ 𝑛 must lie in the set 𝑋 𝑤 = 𝐵 produced by the entire circuit. Thisobservation motivates to associate with every gate 𝑣 its residue 𝑌 𝑣 = { 𝑦 ∈ ℕ 𝑛 ∶ 𝑋 𝑣 + 𝑦 ⊆ 𝐵 } which collects all vectors 𝑦 ∈ ℕ 𝑛 , the translates of 𝑋 𝑣 by which lie in the set 𝐵 produced by the entirecircuit. For example, if 𝑣 is the output gate, then 𝑋 𝑣 = 𝐵 and 𝑌 𝑣 = { ⃗ . If 𝑣 is an input node, theneither 𝑋 𝑣 = { ⃗ and 𝑌 𝑣 = 𝐵 , or 𝑋 𝑣 = { ⃗𝑒 𝑖 } and 𝑌 𝑣 = { 𝑏 − ⃗𝑒 𝑖 ∶ 𝑏 ∈ 𝐵, 𝑏 𝑖 ⩾ .Note that neither 𝑋 𝑣 nor 𝑌 𝑣 needs lie in 𝐵 , but 𝑋 𝑣 + 𝑌 𝑣 ⊆ 𝐵 already holds for every gate 𝑣 . Thus,if the circuit Φ has 𝑡 gates, then we obtain a covering of the set 𝐵 by 𝑡 sumsets of the form 𝑋 𝑣 + 𝑌 𝑣 A norm-measure is any assignment 𝜇 ∶ ℕ 𝑛 → ℝ + of nonnegative real numbers to vectors in ℕ 𝑛 such that every - vector with at most one gets norm at most , and the norm is monotone andsubadditive: 𝜇 ( 𝑥 ) ⩽ 𝜇 ( 𝑥 + 𝑦 ) ⩽ 𝜇 ( 𝑥 ) + 𝜇 ( 𝑦 ) holds for all vectors 𝑥, 𝑦 ∈ ℕ 𝑛 . Lemma 5.6 (Decomposition lemma [16]) . If a set
𝐵 ⊂ ℕ 𝑛 can be produced by a Minkowski (∪ , +) circuit of size 𝑡 , then 𝐵 is a union of 𝑡 sumsets 𝑋 + 𝑌 ⊆ 𝐵 with the following property. (∗)
For every norm-measure 𝜇 ∶ ℕ 𝑛 → ℝ + , for every vector 𝑏 ∈ 𝐵 of norm 𝜇 ( 𝑏 ) > , and every 𝜇 ( 𝑏 ) ⩽ 𝜃 < at least one of these sumsets 𝑋 + 𝑌 contains vectors 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 suchthat 𝑥 + 𝑦 = 𝑏 and 𝜃 ⋅ 𝜇 ( 𝑏 ) < 𝜇 ( 𝑥 ) ⩽ 𝜃 ⋅ 𝜇 ( 𝑏 ) . The lemma was originally proved in [16, Theorem D]. Here we give a simpler proof.
Proof.
Let Φ be a Minkowski (∪ , +) circuit of size 𝑡 producing the set 𝐵 . Since we have only 𝑡 gatesin the circuit, it is enough to show that the collection of sumsets 𝑋 𝑣 + 𝑌 𝑣 associated with the gates 𝑣 of Φ has the desired property (∗) . So, fix some norm-measure 𝜇 ∶ ℕ 𝑛 → ℝ + , some vector 𝑏 ∈ 𝐵 ofnorm 𝑝 ∶= 𝜇 ( 𝑏 ) > , and a real number 𝑝 ⩽ 𝜃 < .By a decomposition of the vector 𝑏 (or just a decomposition , because the vector 𝑏 is fixed) at a gate 𝑣 we will mean a pair ( 𝑥, 𝑦 ) ∈ 𝑋 𝑣 × 𝑌 𝑣 of vectors (if there is one) such that 𝑥 + 𝑦 = 𝑏 . The norm ofsuch a decomposition is the norm 𝜇 ( 𝑥 ) of the first vector (that in the set 𝑋 𝑣 ). Note that at the outputgate, we have the unique decomposition ( 𝑥, 𝑦 ) = ( 𝑏, ⃗ of 𝑏 of norm 𝜇 ( 𝑥 ) = 𝜇 ( 𝑏 ) = 𝑝 . Claim.
Let 𝑣 be a gate entered from gates 𝑢 and 𝑤 . If there is a decomposition ( 𝑥, 𝑦 ) of vector 𝑏 atgate 𝑣 , then there is a decomposition ( 𝑥 ′ , 𝑦 ′ ) of 𝑏 at 𝑢 or 𝑤 such that ⋅ 𝜇 ( 𝑥 ) ⩽ 𝜇 ( 𝑥 ′ ) ⩽ 𝜇 ( 𝑥 ) .Proof. If 𝑣 = 𝑢 ∪ 𝑤 is a union gate, then 𝑋 𝑣 = 𝑋 𝑢 ∪ 𝑋 𝑤 and, hence, 𝑌 𝑣 = 𝑌 𝑢 ∩ 𝑌 𝑤 . So, the same pair ( 𝑥, 𝑦 ) is a decomposition at the gate 𝑢 (if 𝑥 ∈ 𝑋 𝑢 ) or at the gate 𝑤 (if 𝑥 ∈ 𝑋 𝑤 ), and the claim is trivialin this case.Assume now that 𝑣 = 𝑢 + 𝑤 is a Minkowski sum gate. Then 𝑥 = 𝑥 𝑢 + 𝑥 𝑤 for some vectors 𝑥 𝑢 ∈ 𝑋 𝑢 and 𝑥 𝑤 ∈ 𝑋 𝑤 . Since vector 𝑦 belongs to the residue 𝑌 𝑣 of gate 𝑣 , we know that 𝑋 𝑢 + 𝑋 𝑤 + 𝑦 ⊆ 𝐵 holds. In particular, both inclusions 𝑋 𝑢 + ( 𝑥 𝑤 + 𝑦 ) ⊆ 𝐵 and 𝑋 𝑤 + ( 𝑥 𝑢 + 𝑦 ) ⊆ 𝐵 must hold. So, vector 𝑥 𝑤 + 𝑦 belongs to the residue 𝑌 𝑢 of gate 𝑢 , and vector 𝑥 𝑢 + 𝑦 belongs to the residue 𝑌 𝑤 of gate 𝑤 . Thisimplies that the pair ( 𝑥 𝑢 , 𝑥 𝑤 + 𝑦 ) is a decomposition of 𝑏 at the gate 𝑢 , and the pair ( 𝑥 𝑤 , 𝑥 𝑢 + 𝑦 ) is a decomposition of 𝑏 at the gate 𝑤 . Since 𝑥 = 𝑥 𝑢 + 𝑥 𝑤 , the monotonicity of the norm implies that both 𝜇 ( 𝑥 𝑢 ) and 𝜇 ( 𝑥 𝑤 ) are at most 𝜇 ( 𝑥 ) , while the subadditivity of the norm implies that one of the norms 𝜇 ( 𝑥 𝑢 ) and 𝜇 ( 𝑥 𝑤 ) of these decompositions must be at least ⋅ 𝜇 ( 𝑥 𝑢 + 𝑥 𝑤 ) = ⋅ 𝜇 ( 𝑥 ) , and we can takethat input 𝑢 or 𝑤 at which the decomposition has larger norm. (cid:3) We now start at the output gate with the unique decomposition ( 𝑥, 𝑦 ) = ( 𝑏, ⃗ of vector 𝑏 , andtraverse an input-output path 𝑃 in the circuit backwards by using the following rule: if 𝑣 is a currentlyreached gate, and ( 𝑥, 𝑦 ) is a decomposition at this gate, then go to that of the two inputs of 𝑣 whichhas a decomposition ( 𝑥 ′ , 𝑦 ′ ) of norm 𝜇 ( 𝑥 ′ ) ⩾ ⋅ 𝜇 ( 𝑥 ) (if both input gates have this property, then goto any of them). The claim above ensures that we will eventually reach some input node.If this input node holds the set { ⃗ , then the only decomposition ( 𝑥, 𝑦 ) = ( ⃗ , 𝑏 ) of vector 𝑏 at this gatehas norm 𝜇 ( 𝑥 ) = 𝜇 ( ⃗ ⩽ , and if this gate holds { ⃗𝑒 𝑖 } , then the only decomposition ( 𝑥, 𝑦 ) = ( ⃗𝑒 𝑖 , 𝑏 − ⃗𝑒 𝑖 ) of 𝑏 at this gate has also norm 𝜇 ( 𝑥 ) = 𝜇 ( ⃗𝑒 𝑖 ) ⩽ . In both cases, we have that 𝜇 ( 𝑥 ) ⩽ , which is atmost 𝜃𝑝 , because 𝜃 ⩾ 𝑝 .On the other hand, the (also unique) decomposition ( 𝑥, 𝑦 ) = ( 𝑏, ⃗ of the vector 𝑏 at the outputgate has norm 𝜇 ( 𝑥 ) = 𝜇 ( 𝑏 ) = 𝑝 , which is strictly larger than 𝜃𝑝 , because 𝜃 < . So, there must bean edge ( 𝑢, 𝑣 ) in the path 𝑃 at which the jump from ⩽ 𝜃𝑝 to > 𝜃𝑝 happens. That is, there must be adecomposition ( 𝑥, 𝑦 ) at the gate 𝑣 and a decomposition ( 𝑥 ′ , 𝑦 ′ ) at the gate 𝑢 such that 𝜇 ( 𝑥 ) > 𝜃𝑝 but 𝜇 ( 𝑥 ′ ) ⩽ 𝜃𝑝 . By the above claim, we have 𝜇 ( 𝑥 ′ ) ⩾ ⋅ 𝜇 ( 𝑥 ) . We have thus found a sumset 𝑋 𝑢 + 𝑌 𝑢 andvectors 𝑥 ′ ∈ 𝑋 𝑢 and 𝑦 ′ ∈ 𝑌 𝑢 such that 𝑥 ′ + 𝑦 ′ = 𝑏 and 𝜃𝑝 < 𝜇 ( 𝑥 ′ ) ⩽ 𝜃𝑝 , as desired. (cid:3) Decomposition lemma for approximating (max , +) circuits. In the following lemma, < 𝛾 < is an arbitrary fixed “balance” parameter. For a - vector 𝑎 , let | 𝑎 | ∶= ⟨ 𝑎, 𝑎 ⟩ denote the number of s in 𝑎 . Lemma 5.7.
Let Φ be a (max , +) circuit of size 𝑡 approximating the maximization problem on a set 𝐴 ⊆ {0 , 𝑛 within factor 𝑟 ⩾ , and let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by Φ . Then there exist 𝑡 or fewer sumsets 𝑋 + 𝑌 ⊆ 𝐵 such that: (1) if 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 , then 𝑥 + 𝑦 ⩽ 𝑎 for some 𝑎 ∈ 𝐴 ; (2) ⟨ 𝑥, 𝑦 ⟩ = 0 holds for all 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 ; (3) for every vector 𝑎 ∈ 𝐴 with | 𝑎 | ⩾ 𝑟 ∕ 𝛾 ones, at least one of these sumsets 𝑋 + 𝑌 containsvectors 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 such that: (5) ⟨ 𝑎, 𝑥 + 𝑦 ⟩ ⩾ 𝑟 ⋅ | 𝑎 | , ⟨ 𝑎, 𝑥 ⟩ > 𝛾 𝑟 ⋅ | 𝑎 | and ⟨ 𝑎, 𝑦 ⟩ ⩾ 𝛾𝑟 ⋅ | 𝑎 | . Proof.
By Lemma 5.5, we know that the set 𝐵 has the following two properties:(i) if 𝑏 ∈ 𝐵 , then 𝑏 ⩽ 𝑎 for some 𝑎 ∈ 𝐴 ;(ii) if 𝑎 ∈ 𝐴 , then ⟨ 𝑎, 𝑏 ⟩ ⩾ 𝑟 ⋅ | 𝑎 | for some 𝑏 ∈ 𝐵 .The Minkowski (∪ , +) version Φ ′ of the circuit Φ has the same size 𝑡 and produces the same set 𝐵 .When applied to the Minkowski circuit Φ ′ , Lemma 5.6 gives us a collection of 𝑡 sumsets 𝑋 + 𝑌 ⊆ 𝐵 with the following property holding for every norm-measure 𝜇 ∶ ℕ 𝑛 → ℝ + , for every vector 𝑏 ∈ 𝐵 ofnorm 𝜇 ( 𝑏 ) > , and every real number 𝜃 satisfying 𝜇 ( 𝑏 ) ⩽ 𝜃 < : (∗) at least one of the sumsets 𝑋 + 𝑌 contains vectors 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 such that 𝑥 + 𝑦 = 𝑏 and 𝜃 ⋅ 𝜇 ( 𝑏 ) < 𝜇 ( 𝑥 ) ⩽ 𝜃 ⋅ 𝜇 ( 𝑏 ) .Since the set 𝐴 consists of only - vectors, property (i) implies that the set 𝐵 also consists of - vectors. So, 𝑋 + 𝑌 ⊆ 𝐵 implies that each of our sumsets 𝑋 + 𝑌 has the first two properties (1) and(2) claimed in Lemma 5.7. It remains to show the third “balancedness” property (3). Fix an arbitrary vector 𝑎 ∈ 𝐴 with | 𝑎 | ⩾ 𝑟 ∕ 𝛾 ones. Property (ii) of the set 𝐵 suggests to associatewith 𝑎 the norm-measure 𝜇 ( 𝑥 ) = 𝜇 𝑎 ( 𝑥 ) ∶= ⟨ 𝑎, 𝑥 ⟩ . Then, by (ii), there is a vector 𝑏 ∈ 𝐵 of norm 𝜇 ( 𝑏 ) = ⟨ 𝑎, 𝑏 ⟩ ⩾ 𝑝 ∶= | 𝑎 | ∕ 𝑟 , which is > since | 𝑎 | ⩾ 𝑟 ∕ 𝛾 , 𝑟 ⩾ and 𝛾 < . We also have 𝛾 ⩾ 𝑟 ∕ | 𝑎 | ⩾ 𝜇 ( 𝑏 ) . Hence, by the property (∗) , at least one of our sumsets 𝑋 + 𝑌 contains vectors 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 such that 𝑥 + 𝑦 = 𝑏 and the following inequalities hold:(6) 𝛾 ⋅ 𝑝 < 𝜇 ( 𝑥 ) = ⟨ 𝑎, 𝑥 ⟩ ⩽ 𝛾 ⋅ 𝑝 . Now, the first inequality in Eq. (5) follows since ⟨ 𝑎, 𝑥 + 𝑦 ⟩ = ⟨ 𝑎, 𝑏 ⟩ ⩾ 𝑝 , the second inequality inEq. (5) is the first inequality in Eq. (6) and, since ⟨ 𝑥, 𝑦 ⟩ = 0 , the third inequality in Eq. (5) followsfrom ⟨ 𝑎, 𝑥 + 𝑦 ⟩ ⩾ 𝑝 and the second inequality in Eq. (6). (cid:3) The rectangle bound. A rectangle is a family of sets specified by giving a pair A , B of familiesof sets which is cross-disjoint in that 𝐴 ∩ 𝐵 = ∅ holds for all sets 𝐴 ∈ A and 𝐵 ∈ B . The rectangle R = A ∨ B itself consists of all sets 𝐴 ∪ 𝐵 with 𝐴 ∈ A and 𝐵 ∈ B . A rectangle R lies below a family F if every set of R is contained in at least one set of F .For real numbers 𝑟 ⩾ and < 𝛾 < , we say that a set 𝐹 appears ( 𝑟, 𝛾 ) -balanced in a rectangle R = A ∨ B if there are sets 𝐴 ∈ A and 𝐵 ∈ B such that(7) | 𝐹 ∩ ( 𝐴 ∪ 𝐵 ) | ⩾ 𝑟 ⋅ | 𝐹 | , | 𝐹 ∩ 𝐴 | > 𝛾 𝑟 ⋅ | 𝐹 | and | 𝐹 ∩ 𝐵 | ⩾ 𝛾𝑟 ⋅ | 𝐹 | . Thus, a set 𝐹 appears balanced in a rectangle R , if for some pair of disjoint sets whose union belongs to R , the union as well as the sets themselves contain “many enough” elements of 𝐹 . In our applications,we will only use the last two inequalities in Eq. (7), but the first inequality may also be important insome applications.The following lemma is just a translation of Lemma 5.7 from the language of vectors to the lan-guage of sets. Recall that 𝖬𝖺𝗑 𝑟 ( F ) denotes the minimum size of a (max , +) circuit approximating themaximization problem on F within the factor 𝑟 . Lemma 5.8.
Let 𝑟 ⩾ and < 𝛾 < . If 𝖬𝖺𝗑 𝑟 ( F ) ⩽ 𝑡 , then there are 𝑡 or fewer rectangles lying below F such that every set of F with at least 𝑟 ∕ 𝛾 elements appears ( 𝑟, 𝛾 ) -balanced in at least one of theserectangles.Proof. Take a (max , +) circuit Φ of size 𝑡 = 𝖬𝖺𝗑 𝑟 ( F ) approximating the maximization problem on F within factor 𝑟 . Let 𝐴 ⊆ {0 , 𝑛 be the set of characteristic - vectors of the sets in F . Hence, thecircuit 𝑟 -approximates the maximization problem on 𝐴 . Let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by Φ .Lemma 5.7 gives us 𝑡 or fewer sumsets 𝑋 + 𝑌 ⊆ 𝐵 with properties (1)–(3) listed in this lemma.By property (1), the set 𝐵 and, hence, each of our 𝑡 sumsets 𝑋 + 𝑌 consists of only - vectors.So, each sumset 𝑋 + 𝑌 translates into the rectangle R = A ∨ B , where A is the family of supports 𝑆 𝑥 = { 𝑖 ∶ 𝑥 𝑖 = 1} of vectors 𝑥 ∈ 𝑋 , and B is the family of supports 𝑆 𝑦 = { 𝑖 ∶ 𝑦 𝑖 = 1} of vectors 𝑦 ∈ 𝑌 .After this translation, property (1) of Lemma 5.7 implies that each of these rectangles R = A ∨ B liesbelow our family F , property (2) yields the cross-disjointness condition ( 𝐴 ∩ 𝐵 = ∅ for all sets 𝐴 ∈ A and 𝐵 ∈ B ), and property (3) implies that every set 𝐹 ∈ F with | 𝐹 | ⩾ 𝑟 ∕ 𝛾 elements appears ( 𝑟, 𝛾 ) -balanced in at least one of these rectangles. (cid:3) In applications, we will use a direct consequence of Lemma 5.8 formulated as a lower bound on 𝖬𝖺𝗑 𝑟 ( F ) . Let 𝑟 ⩾ be a given approximation factor, and < 𝛾 < any fixed real parameter (we arefree to choose this parameter). Theorem 5.9 (Rectangle bound) . Let F be a family of feasible solutions, and H ⊆ F some subfamilyof feasible solutions, each of cardinality at least 𝑟 ∕ 𝛾 . If for every rectangle R lying below F , at most ℎ of the sets of H appear ( 𝑟, 𝛾 ) -balanced, then 𝖬𝖺𝗑 𝑟 ( F ) ⩾ | H | ∕ ℎ . In particular, Result 4 stated in Section 2 corresponds to the balance parameter 𝛾 = 2∕3 . In thefollowing applications, we will always take F = H , but the possibility to choose special subfamilies H ⊆ F of feasible solutions may be useful in other applications. Note that, besides of the cross-disjointness and balancedness of rectangles, the fact that the rectangles must lie below F is also impor-tant. If, say, F is the family of all perfect matchings in some graph, and if a rectangle R lies below F ,then we immediately know that every set of R must be a union of two vertex-disjoint matchings.5.7. Maximization on designs.
We already know that there exist many maximization problems forwhich slight decrease of the allowed approximation factor from 𝑟 = 1 + 𝑜 (1) to 𝑟 = 1 can exponentiallyincrease the size of (max , +) circuits (Proposition 5.2). We also know explicit maximization problemsfor which such a jump in circuit size occurs when decreasing the approximation factor from 𝑟 = 2 to 𝑟 = 1 (Theorem 5.4).Our goal in this section is to show that such jumps can happen for arbitrarily large approximation fac-tors 𝑟 : a slight decrease of the allowed approximation factor 𝑟 can make tractable problems intractable.We demonstrate these jumps on maximization problems whose families F of feasible solutions are“combinatorial designs” (Theorem 5.12 below).An ( 𝑚, 𝑑 ) - design (know also as a weak design ) is a family F which is: ◦ 𝑚 - uniform : every set has exactly 𝑚 elements; ◦ 𝑑 - disjoint : no two distinct sets share 𝑑 or more elements in common.We will see soon (the upper bound in Theorem 5.12) that, for some ( 𝑚, 𝑑 ) -designs F ⊆ [ 𝑛 ] with 𝑛 = 𝑚 , 𝖬𝖺𝗑 𝑟 ( F ) = 𝑂 ( 𝑛 ) holds when the approximation factor 𝑟 = 𝑚 ∕ 𝑑 is allowed. On the other hand,we have the following general lower bound on 𝖬𝖺𝗑 𝑟 ( F ) for every ( 𝑚, 𝑑 ) design F , when the allowedapproximation factor 𝑟 is only slightly smaller than 𝑚 ∕ 𝑑 .To state this bound, we need an auxiliary notation. For a family F of sets and a real number 𝑙 ⩾ ,let 𝑙 ( F ) denote the maximal possible number of sets in F containing a fixed set with 𝑙 (or more)elements. In other words, 𝑙 ( F ) is the maximal possible number of sets in F whose intersection has 𝑙 (or more) elements. In particular, if 𝑚 is the maximum cardinality of a set of F , then | F | = ( F ) ⩾ ( F ) ⩾ … ⩾ 𝑚 ( F ) = 1 , and 𝑙 ( F ) = 0 for all 𝑙 > 𝑚 . Note that a nonempty 𝑚 -uniform family F is an ( 𝑚, 𝑑 ) -design if and only if 𝑑 ( F ) = 1 . Also, ( F ) = 1 means that all sets of F are disjoint. Lemma 5.10.
Let F be an ( 𝑚, 𝑑 ) -design for ⩽ 𝑑 < 𝑚 , 𝑑 + 1) ⩽ 𝛾 < and 𝑙 = 𝛾𝑑 ∕2 . For thefactor 𝑟 = (1 − 𝛾 ) 𝑚 ∕ 𝑑 , we have 𝖬𝖺𝗑 𝑟 ( F ) ⩾ | F | 𝑙 ( F ) . Proof.
We are going to apply the rectangle bound (Theorem 5.9) with the balance parameter 𝛾 . First,observe that | 𝐹 | ⩾ 𝑟 ∕ 𝛾 holds for every set 𝐹 ∈ F : since | 𝐹 | = 𝑚 , this is equivalent to the inequality ⩾ (1 − 𝛾 )∕ 𝛾𝑑 , which holds because 𝛾 ⩾ 𝑑 + 1) .Take an arbitrary rectangle R = A ∨ B lying below F . Let F R ⊆ F be the family of all sets 𝐹 ∈ F such that | 𝐹 ∩ 𝐴 | ⩾ 𝛾 𝑟 ⋅ 𝑚 = 𝛾 𝛾 ) ⋅ 𝑑 ⩾ 𝛾 ⋅ 𝑑 = 𝑙 and | 𝐹 ∩ 𝐵 | ⩾ 𝛾𝑟 ⋅ 𝑚 = 𝑑 hold for some 𝐴 ∈ A and 𝐵 ∈ B . By Theorem 5.9, it is enough to show that | F R | ⩽ 𝑙 ( F ) . Wecan assume that all sets 𝐵 ∈ B have | 𝐵 | ⩾ 𝑑 elements: sets 𝐵 ∈ B of size | 𝐵 | < 𝑑 cannot fulfill | 𝐹 ∩ 𝐵 | ⩾ 𝑑 and, hence, can be removed from B without changing F R . Similarly, we can assume thatall sets 𝐴 ∈ A have | 𝐴 | ⩾ 𝑙 elements. Let 𝑋 be the union of all sets in A ; hence, | 𝑋 | ⩾ 𝑙 . Claim.
All sets of F R contain the set 𝑋 .Proof. Take a set 𝐹 ∈ F R . Then | 𝐹 ∩ 𝐵 | ⩾ 𝑑 holds for some set 𝐵 ∈ B . On the other hand, since therectangle R lies below F , every set 𝐴 ∪ 𝐵 with 𝐴 ∈ A must lie in some set of F . Since all these setscontain the set 𝐵 with | 𝐵 | ⩾ 𝑑 elements, and since the family F is 𝑑 -disjoint, this implies that all setsof A ∨ { 𝐵 } and, hence, also the set 𝑋 ∪ 𝐵 must be contained in one set 𝐹 𝐵 of F . Since both sets 𝐹 and 𝐹 𝐵 of F contain the same set 𝐹 ∩ 𝐵 of size | 𝐹 ∩ 𝐵 | ⩾ 𝑑 , and since the family F is 𝑑 -disjoint, theequality 𝐹 = 𝐹 𝐵 and, hence, the desired inclusion 𝑋 ⊆ 𝐹 follows. (cid:3)
Since the set 𝑋 has | 𝑋 | ⩾ 𝑙 elements, the claim yields | F R | ⩽ | 𝑋 | ( F ) ⩽ 𝑙 ( F ) and, by Theorem 5.9(applied with H = F ), the desired lower bound 𝖬𝖺𝗑 𝑟 ( F ) ⩾ | F | ∕ | F R | ⩾ | F | ∕ 𝑙 ( F ) follows. (cid:3) Factor hierarchy theorem.
We will now apply the general lower bound of Lemma 5.10 to ex-plicit designs. Let 𝑚 be a prime power, ⩽ 𝑑 ⩽ 𝑚 an integer, and consider the grid GF( 𝑚 ) × GF( 𝑚 ) .The polynomial ( 𝑚, 𝑑 ) -design F consists of all | F | = 𝑚 𝑑 subsets 𝑆 of points in this grid of the form 𝑆 = {( 𝑎, 𝑝 ( 𝑎 )) ∶ 𝑎 ∈ GF( 𝑚 )} for a univariate polynomial 𝑝 = 𝑝 ( 𝑥 ) of degree at most 𝑑 − 1 over GF( 𝑚 ) .Note that no two points of any of these sets 𝑆 lie in the same row of the grid.The main combinatorial property of polynomial designs is the following. Proposition 5.11.
Let F be a polynomial ( 𝑚, 𝑑 ) -design, and ⩽ 𝑑 ⩽ 𝑚 . For every set of 𝑙 ⩽ 𝑑 pointsof the grid GF( 𝑚 ) × GF( 𝑚 ) , with no two in the same row, exactly 𝑚 𝑑 − 𝑙 sets of F contain this set. Inparticular, 𝑙 ( F ) ⩽ 𝑚 𝑑 − 𝑙 holds for every ⩽ 𝑙 ⩽ 𝑑 .Proof. This is a direct consequence of a standard result in polynomial interpolation. For any 𝑙 ⩽ 𝑑 distinct points ( 𝑎 , 𝑏 ) , … , ( 𝑎 𝑙 , 𝑏 𝑙 ) in GF( 𝑚 )×GF( 𝑚 ) , the number of polynomials 𝑝 ( 𝑥 ) of degree at most 𝑑 −1 satisfying 𝑝 ( 𝑎 ) = 𝑏 , … , 𝑝 ( 𝑎 𝑙 ) = 𝑏 𝑙 is either (if 𝑎 𝑖 = 𝑎 𝑗 holds for some 𝑖 ≠ 𝑗 ) or is exactly 𝑚 𝑑 − 𝑙 :this latter number is exactly the number of solutions of the corresponding system of linear equations,with coefficients of 𝑝 viewed as variables. (cid:3) We already know that (min , +) circuits approximating minimization problems on polynomial de-signs (within any finite factor 𝑟 = 𝑟 ( 𝑚 ) ) must be large (Corollary 4.3). Now we show that also the maximization problem on the polynomial ( 𝑚, 𝑑 ) -design is hard to approximate, as long as the allowedapproximation factor is smaller than 𝑚 ∕ 𝑑 . Theorem 5.12 (Factor hierarchy theorem) . Let F be a polynomial ( 𝑚, 𝑑 ) -design for ⩽ 𝑑 < 𝑚 , and 𝑑 + 1) ⩽ 𝛾 < . Then: (i) 𝖬𝖺𝗑 𝑟 ( F ) ⩽ 𝑚 for the factor 𝑟 = 𝑚 ∕ 𝑑 , but (ii) 𝖬𝖺𝗑 𝑟 ( F ) ⩾ 𝑚 𝛾𝑑 ∕2 for the factor 𝑟 = (1 − 𝛾 ) 𝑚 ∕ 𝑑 . That is, the maximization problem on F can be approximated by a small (max , +) circuit within thefactor 𝑟 = 𝑚 ∕ 𝑑 , but the size of approximating (max , +) circuits drastically increases when this factoris only slightly decreased. Proof.
To show the lower bound (ii), we just apply Lemma 5.10, which yields the lower bound
𝖬𝖺𝗑 𝑟 ( F ) ⩾ | F | ∕ 𝑙 ( F ) for 𝑙 = 𝛾𝑑 ∕2 . Since | F | = 𝑚 𝑑 and, by Proposition 5.11, 𝑙 ( F ) ⩽ 𝑚 𝑑 − 𝑙 , the desired lowerbound 𝖬𝖺𝗑 𝑟 ( F ) ⩾ 𝑚 𝑙 follows.So, it remains to show the upper bound (i). Given an input weighting 𝑥 of the points of the grid GF( 𝑚 ) × GF( 𝑚 ) , we can first use 𝑚 ( 𝑚 − 1) max operations to compute 𝑚 numbers 𝑦 , … , 𝑦 𝑚 , where 𝑦 𝑖 is the maximum weight of a point in the 𝑖 th row of the grid. We then apply the (max , +) circuit for the top 𝑑 -of- 𝑚 selection problem (see Proposition 5.1) to compute the sum 𝑊 of the largest 𝑑 of thenumbers 𝑦 , … , 𝑦 𝑚 using at most 𝑑𝑚 additional (max , +) operations. Hence, 𝑊 is a sum of weightsof 𝑑 heaviest points in the grid with no two in the same row: each 𝑦 𝑖 picks only one point in the 𝑖 th row.The main combinatorial property of polynomial designs (Proposition 5.11) implies that these 𝑑 pointsare contained in a (unique) set of F . Hence, the found value 𝑊 cannot exceed the optimal value (theweights are nonnegative). On the other hand, the weight of 𝑑 heaviest points of an optimal solution 𝐹 ∈ F cannot exceed 𝑊 . Since | 𝐹 | = 𝑚 , the weight of this solution cannot exceed ( 𝑚 ∕ 𝑑 ) 𝑊 , asdesired. (cid:3) Remark . The maximizing greedy algorithm also achieves the same approximation factor 𝑚 ∕ 𝑑 on thepolynomial ( 𝑚, 𝑑 ) -design: it will also first take the heaviest 𝑑 points of the grid GF( 𝑚 ) × GF( 𝑚 ) , withno two lying in the same row. But this is already the best the greedy algorithm can do.To show this, take 𝜖 > arbitrarily small, and set 𝑐 ∶= 1∕(1 − 𝜖 ∕2) > . Take arbitrary two sets 𝐴 ≠ 𝐵 ∈ F , and a subset 𝑆 ⊂ 𝐴 of | 𝑆 | = 𝑑 elements. Since F is an ( 𝑚, 𝑑 ) -design, 𝑆 cannot becontained in 𝐵 . So, give weight 𝑐 > to all elements of 𝑆 , weight to all elements of 𝐵 ⧵ 𝑆 , andzero weight to the rest. Then the maximizing (best-in) greedy algorithm picks elements of weight 𝑐 first, gets all | 𝑆 | = 𝑑 of them, but then is stuck because no element of weight fits; hence, the greedyalgorithm achieves the total weight 𝑐 | 𝑆 | = 𝑐𝑑 . But the optimum is at least | 𝐵 | = 𝑚 . Hence, theapproximation factor is at least 𝑚 ∕ 𝑐𝑑 = (1 − 𝜖 ∕2) 𝑚 ∕ 𝑑 > (1 − 𝜖 ) 𝑚 ∕ 𝑑 .5.9. Greedy can beat approximating (max , +) circuits. As Remark 6 shows, Theorem 5.12 does not imply that the maximizing greedy algorithm can beat approximating (max , +) circuits: small (max , +) circuits can also achieve the greedy approximation factor on designs.To show that the greedy algorithm can still outperform approximating (max , +) circuits, we consideranother maximization problem: maximum weight matchings in 𝑘 -partite 𝑘 -uniform hypergraphs. Wehave a set 𝑉 = 𝑉 ∪ ⋯ ∪ 𝑉 𝑘 of | 𝑉 | = 𝑚𝑘 vertices decomposed into 𝑘 disjoint blocks 𝑉 , … , 𝑉 𝑘 , eachof size 𝑚 . Edges (called also hyperedges ) are 𝑘 -tuples 𝑒 ∈ 𝑉 × ⋯ × 𝑉 𝑘 . The ground set 𝐸 consistsof all | 𝐸 | = 𝑚 𝑘 edges. Two edges are disjoint if they differ in all 𝑘 positions. A matching is a set ofdisjoint edges, and is a perfect matching if it has the maximum possible number 𝑚 of edges.The family F 𝑚,𝑘 of feasible solutions of our problem consists of all | F 𝑚,𝑘 | = ( 𝑚 !) 𝑘 −1 perfect match-ings. So, the maximization problem on F 𝑚,𝑘 is, given an assignment of nonnegative weights 𝑥 𝑒 to theedges 𝑒 ∈ 𝐸 , to compute the maximum total weight 𝑓 ( 𝑥 ) = max { 𝑥 𝑒 + ⋯ + 𝑥 𝑒 𝑚 ∶ 𝑒 𝑖 ∈ 𝐸 , and 𝑒 𝑖 and 𝑒 𝑗 are disjoint for all 𝑖 ≠ 𝑗 } of a perfect matching. Note that in the case 𝑘 = 2 , F 𝑚,𝑘 consists of perfect matchings in 𝐾 𝑚,𝑚 , and theproblem is to compute the maximum weight of such a perfect matching.The greedy algorithm can approximate the maximization problem on F 𝑚,𝑘 within the factor 𝑘 byjust always picking the heaviest of the remaining edges, untouched by the partial matching picked sofar. On the other hand, we have the following lower bound for (max , +) circuits approximating thisproblem. Theorem 5.13.
Let 𝑚 be a sufficiently large integer, and 𝑘 = 𝑘 ( 𝑚 ) be an integer such that ⩽ 𝑘 ⩽ log √ 𝑚 . If 𝑟 ⩽ 𝑘 ∕9 , then 𝖬𝖺𝗑 𝑟 ( F 𝑚,𝑘 ) = 2 Ω( √ 𝑚 ) . Proof.
We are going to apply the rectangle bound (Theorem 5.9) with 𝛾 ∶= 2∕3 ; note that then both 𝛾 and 𝛾 in Eq. (7) are equal to . So, take an arbitrary rectangle R = A ∨ B lying below F = F 𝑚,𝑘 .Hence, sets in A and in B are subsets of (hyper-)edges 𝑒 ∈ 𝑉 × ⋯ × 𝑉 𝑘 . Since R lies below our family F , and F consist of (perfect) matchings, all sets 𝐴 ∪ 𝐵 with 𝐴 ∈ A and 𝐵 ∈ B must also be matchings.Take the integer 𝑑 ∶= ⌈ 𝑚 ∕3 𝑟 ⌉ , and consider the family F R = { 𝐹 ∈ F ∶ | 𝐹 ∩ 𝐴 | ⩾ 𝑑 and | 𝐹 ∩ 𝐵 | ⩾ 𝑑 for some 𝐴 ∈ A and 𝐵 ∈ B } . Our goal is to show a possibly small upper bound | F R | ⩽ ℎ on the number of sets in any such family.Then Theorem 5.9 (for the balance parameter 𝛾 = 2∕3 ) gives the lower bound 𝖬𝖺𝗑 𝑟 ( F ) ⩾ | F | ∕ ℎ .Since the rectangle R = A ∨ B is cross-disjoint, we know that the matchings 𝐴 ∈ A and 𝐵 ∈ B must be edge-disjoint, that is, 𝐴 ∩ 𝐵 = ∅ must hold. However, since the sets 𝐴 ∪ 𝐵 are also matchings( R lies below F ), we actually know that matchings 𝐴 and 𝐵 are even vertex-disjoint : if 𝑆 ⊆ 𝑉 is theset of vertices belonging to at least one edge of a matching in A , and 𝑇 ⊆ 𝑉 is the set of verticesbelonging to at least one edge of a matching in B , then 𝑆 ∩ 𝑇 = ∅ (this is a crucial property). Note thatin the proof of the lower bound for designs (Lemma 5.10), the cross-disjointness property of rectangleswas not used.So, call a matching 𝐴 ⊆ 𝑉 × ⋯ × 𝑉 𝑘 an 𝑆 - matching if 𝐴 ⊆ ( 𝑉 ∩ 𝑆 ) × ⋯ × ( 𝑉 𝑘 ∩ 𝑆 ) holds, thatis, if edges of 𝐴 only match vertices of 𝑆 ; 𝑇 - matchings are defined similarly. By the definition of F R ,every perfect matching 𝐹 ∈ F R has at least 𝑑 edges lying in ( 𝑉 ∩ 𝑆 ) × ⋯ × ( 𝑉 𝑘 ∩ 𝑆 ) , and at least 𝑑 edges lying in ( 𝑉 ∩ 𝑇 ) × ⋯ × ( 𝑉 𝑘 ∩ 𝑇 ) . In particular, every perfect matching 𝐹 ∈ F R must containat least one matching 𝐴 ∪ 𝐵 , where 𝐴 is an 𝑆 -matching with | 𝐴 | = 𝑑 edges and 𝐵 is a 𝑇 -matchingwith | 𝐵 | = 𝑑 edges. It therefore suffices to upper-bound the number of perfect matchings 𝐹 with thisproperty.We can pick any such pair ( 𝐴, 𝐵 ) as follows. Let 𝑆 𝑖 = 𝑆 ∩ 𝑉 𝑖 and 𝑇 𝑖 = 𝑇 ∩ 𝑉 𝑖 for 𝑖 = 1 , … , 𝑘 . Wecan assume that each of these 𝑘 sets has at least 𝑑 vertices, for otherwise none of the 𝑆 -matchings orof the 𝑇 -matchings could have ⩾ 𝑑 edges, implying that F R = ∅ . ◦ Pick in each 𝑆 𝑖 a subset 𝑆 ′ 𝑖 ⊆ 𝑆 𝑖 of | 𝑆 ′ 𝑖 | = 𝑑 vertices, and in each 𝑇 𝑖 a subset 𝑇 ′ 𝑖 ⊆ 𝑇 𝑖 of | 𝑇 ′ 𝑖 | = 𝑑 vertices. There are at most 𝑘 ∏ 𝑖 =1 ( 𝑚 𝑖 𝑑 )( 𝑚 − 𝑚 𝑖 𝑑 ) ⩽ ( 𝑚 𝑑 ) 𝑘 possibilities to do this, where 𝑚 𝑖 = | 𝑆 𝑖 | . ◦ Pick a perfect matching 𝐴 in 𝑆 ′1 × ⋯ × 𝑆 ′ 𝑘 and a perfect matching 𝐵 in 𝑇 ′1 × ⋯ × 𝑇 ′ 𝑘 . Thereare only [ ( 𝑑 !) 𝑘 −1 ] = ( 𝑑 !) 𝑘 −1) possibilities to do this.After a pair ( 𝐴, 𝐵 ) of matchings is picked, there are at most [( 𝑚 − 2 𝑑 )!] 𝑘 −1 possibilities to extend 𝐴 ∪ 𝐵 to a perfect matching. Thus, | F R | ⩽ ( 𝑚 𝑑 ) 𝑘 ( 𝑑 !) 𝑘 −1) [( 𝑚 − 2 𝑑 )!] 𝑘 −1 = ( 𝑚 𝑑 ) [ 𝑚 ! ⋅ ( 𝑑𝑑 ) −1 ] 𝑘 −1 , where the equality follows because ( 𝑚 𝑑 ) = 𝑚 !∕(2 𝑑 )!( 𝑚 − 2 𝑑 )! and (2 𝑑 )!∕( 𝑑 !) = ( 𝑑𝑑 ) . Since there are | F | = ( 𝑚 !) 𝑘 −1 perfect matchings, the rectangle bound (Theorem 5.9) yields the following lower boundon 𝑡 = 𝖬𝖺𝗑 𝑟 ( F ) : 𝑡 ⩾ | F || F R | ⩾ ( 𝑑𝑑 ) 𝑘 −1 ( 𝑚 𝑑 ) ⩾ ( 𝑑 𝑑 ) 𝑘 −1 ⋅ ( 𝑑 e 𝑚 ) 𝑑 = 1 𝑑 𝑘 −1 ( 𝑘 𝑑 e 𝑚 ) 𝑑 ⩾ 𝑑 𝑘 −1 ( 𝑘 𝑟 ) 𝑑 , where the second inequality follows from the inequalities ( 𝑚 𝑑 ) ⩽ (e 𝑚 ∕2 𝑑 ) 𝑑 and ( 𝑑𝑑 ) ⩾ 𝑑 ∕ √ 𝑑 ⩾ 𝑑 ∕ 𝑑 , and the last inequality follows because (by our choice) 𝑑 = ⌈ 𝑚 ∕3 𝑟 ⌉ ⩾ 𝑚 ∕3 𝑟 . Our approxima-tion factor is 𝑟 = 2 𝑘 ∕9 . Since clearly 𝑑 ⩽ 𝑚 , we have a lower bound 𝑡 ⩾ ( ) 𝑚 ∕2 𝑘 ⋅ 𝑑 − 𝑘 ⩾ . 𝑚 ∕2 𝑘 − 𝑘 log 𝑚 . From our assumption 𝑘 ⩽ log √ 𝑚 , we have 𝑚 ∕2 𝑘 ⩾ √ 𝑚 ≫ 𝑘 log 𝑚 , and the desired lower bound 𝑡 ⩾ Ω( 𝑚 ∕2 𝑘 ) ⩾ Ω( √ 𝑚 ) follows. (cid:3)
6. W
HAT DO APPROXIMATING TROPICAL CIRCUITS PRODUCE ?If we know that a tropical circuit approximates a given optimization (minimization or maximization)problem within a given factor, what can then be said about the set of vectors produced by that circuit?Using elementary arguments , we partially answered this question in Lemmas 4.2 and 5.5: we gaveproperties, which the sets of produced vectors must necessarily have (these properties were alreadysufficient for our purposes). We will now use convexity arguments to give properties of produced setsthat are also sufficient for circuits to approximate given problems.6.1.
A version of Farkas’ lemma.
Recall that a vector 𝑐 ∈ ℝ 𝑛 is a convex combination (or a weightedaverage ) of vectors ⃗𝑏 , … , ⃗𝑏 𝑚 in ℝ 𝑛 if there are real scalars 𝜆 , … , 𝜆 𝑚 ⩾ such that 𝜆 + ⋯ + 𝜆 𝑚 = 1 and 𝑐 = 𝜆 ⋅ ⃗𝑏 + ⋯ + 𝜆 𝑚 ⋅ ⃗𝑏 𝑚 . It is easy to see the following averaging property : for every vector 𝑥 ∈ ℝ 𝑛 and every convex combi-nation 𝑐 of vectors in 𝐵 , we have min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ ⩽ ⟨ 𝑐, 𝑥 ⟩ ⩽ max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ .We will need the following formulation of Farkas’ lemma due to Fan [8, Theorem 4], see also [27,Corollary 7.1h]. Lemma 6.1 (Farkas’ lemma [8]) . Let 𝑢, ⃗𝑢 , … , ⃗𝑢 𝑚 ∈ ℝ 𝑛 , and 𝛼, 𝛼 , … , 𝛼 𝑚 ∈ ℝ . The following twoassertions are equivalent. (1) ∀ 𝑦 ∈ ℝ 𝑛 inequalities ⟨ ⃗𝑢 , 𝑦 ⟩ ⩾ 𝛼 , … , ⟨ ⃗𝑢 𝑚 , 𝑦 ⟩ ⩾ 𝛼 𝑚 imply ⟨ 𝑢, 𝑦 ⟩ ⩾ 𝛼 . (2) ∃ 𝜆 , … , 𝜆 𝑚 ∈ ℝ + such that 𝑢 = ∑ 𝑖 𝜆 𝑖 ⃗𝑢 𝑖 and 𝛼 ⩽ ∑ 𝑖 𝜆 𝑖 𝛼 𝑖 . This relates optimization with convex combinations.
Lemma 6.2.
For any vectors 𝑎, ⃗𝑎 , … , ⃗𝑎 𝑚 ∈ ℝ 𝑛 the following two assertions are equivalent. (1) ∀ 𝑥 ∈ ℝ 𝑛 + ∶ ⟨ 𝑎, 𝑥 ⟩ ⩾ min 𝑖 ⟨ ⃗𝑎 𝑖 , 𝑥 ⟩ . (2) ∃ 𝜆 , … , 𝜆 𝑚 ∈ ℝ + ∶ ∑ 𝑖 𝜆 𝑖 = 1 and 𝑎 ⩾ ∑ 𝑖 𝜆 𝑖 ⃗𝑎 𝑖 .Proof. The implication (2) ⇒ (1) follows directly from the aforementioned averaging property of con-vex combinations. To show the converse implication (1) ⇒ (2), observe that the assertion (1) is equiv-alent to the assertion that the set of inequalities ⟨ ⃗𝑎 𝑖 , 𝑥 ⟩ ⩾ 𝑧 and ⟨ ⃗𝑒 𝑗 , 𝑥 ⟩ ⩾ for 𝑖 = 1 , … , 𝑚 and 𝑗 = 1 , … , 𝑛 implies the inequality ⟨ 𝑎, 𝑥 ⟩ ⩾ 𝑧 . We use the inequalities ⟨ ⃗𝑒 𝑗 , 𝑥 ⟩ ⩾ to ensure that weonly consider vectors 𝑥 in ℝ 𝑛 + (with no negative entries).By taking 𝑦 = ( 𝑥, 𝑧 ) , 𝑢 = ( 𝑎, −1) , ⃗𝑢 𝑖 = ( ⃗𝑎 𝑖 , −1) for 𝑖 = 1 , … , 𝑚 , and ⃗𝑢 𝑚 + 𝑗 = ( ⃗𝑒 𝑗 , for 𝑗 = 1 , … , 𝑛 ,the above assertion turns into the assertion that for every vector 𝑦 in ℝ 𝑛 +1 , the system of inequalities We use the arrow notation ⃗𝑏 𝑖 for vectors only when they are indexed. ⟨ ⃗𝑢 𝑖 , 𝑦 ⟩ ⩾ for 𝑖 = 1 , … , 𝑚 + 𝑛 , implies the inequality ⟨ 𝑢, 𝑦 ⟩ ⩾ . Then, by Lemma 6.1, there exist 𝜆 , … , 𝜆 𝑚 + 𝑛 ∈ ℝ + such that ( 𝑎, −1) = 𝑚 ∑ 𝑖 =1 𝜆 𝑖 ( ⃗𝑎 𝑖 , −1) + 𝑛 ∑ 𝑗 =1 𝜆 𝑚 + 𝑗 ( ⃗𝑒 𝑗 , . This yields 𝜆 + ⋯ + 𝜆 𝑚 = 1 and 𝑎 ⩾ ∑ 𝑖 𝜆 𝑖 ⃗𝑎 𝑖 , as desired. (cid:3) The following direct consequence of Lemma 6.2 compares the values of optimization problems. Fora set 𝑈 of real vectors, let Conv( 𝑈 ) denote the set of all convex combinations of vectors in 𝑈 , that is,the convex hull of 𝑈 . Say that a set 𝑈 ⊆ ℝ 𝑛 lies above a set 𝑉 ⊆ ℝ 𝑛 if ∀ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑉 ∶ 𝑢 ⩾ 𝑣 , andthat 𝑈 lies below 𝑉 if ∀ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑉 ∶ 𝑢 ⩽ 𝑣 . Lemma 6.3.
Let
𝑈 , 𝑉 ⊂ ℝ 𝑛 be finite sets of vectors. Then (1) ∀ 𝑥 ∈ ℝ 𝑛 + ∶ min 𝑢 ∈ 𝑈 ⟨ 𝑢, 𝑥 ⟩ ⩾ min 𝑣 ∈ 𝑉 ⟨ 𝑣, 𝑥 ⟩ if and only if 𝑈 lies above Conv( 𝑉 ) ; (2) ∀ 𝑥 ∈ ℝ 𝑛 + ∶ max 𝑢 ∈ 𝑈 ⟨ 𝑢, 𝑥 ⟩ ⩽ max 𝑣 ∈ 𝑉 ⟨ 𝑣, 𝑥 ⟩ if and only if 𝑈 lies below Conv( 𝑉 ) . Claim (1) follows directly from Lemma 6.2. The second claim (2) also follows from Lemma 6.2 byusing the equality max( 𝑥, 𝑦 ) = − min(− 𝑥, − 𝑦 ) .6.2. Consequences for tropical circuits.
Recall that the maximization (resp., minimization) problemon a given set
𝐴 ⊂ ℕ 𝑛 of feasible solutions is, for every input weighting 𝑥 ∈ ℝ 𝑛 + , to compute themaximum (resp., minimum) weight ⟨ 𝑎, 𝑥 ⟩ = 𝑎 𝑥 + ⋯ + 𝑎 𝑛 𝑥 𝑛 of a feasible solution 𝑎 ∈ 𝐴 .The consequence of Farkas’ lemma (Lemma 6.3) directly yields the following complete characteri-zation of the properties of sets of vectors produced by approximating (max , +) circuits. Lemma 6.4 (Maximization) . Let
𝐴 ⊂ ℕ 𝑛 be some finite set of vectors, Φ be a (max , +) circuit, and 𝐵 ⊂ ℕ 𝑛 the set of vectors produced by Φ . Then the following two assertions are equivalent. (1) Φ approximates the maximization problem on 𝐴 within a factor 𝑟 . (2) 𝐵 lies below Conv( 𝐴 ) and 𝑟 ⋅ 𝐴 lies below Conv( 𝐵 ) .Proof. By Lemma 3.2, we can assume the circuit Φ is constant-free. Hence, by Proposition 3.1, thecircuit solves the maximization problem of the form Φ( 𝑥 ) = max 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ . The maximization prob-lem on 𝐴 is of the form 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . The circuit approximates the maximization problem on 𝐴 within factor 𝑟 if and only if 𝑟 ⋅ 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) ⩽ 𝑓 ( 𝑥 ) holds for all weightings 𝑥 ∈ ℝ 𝑛 + .When applied with 𝑈 = 𝐵 and 𝑉 = 𝐴 , Lemma 6.3(2) implies that the inequality Φ( 𝑥 ) ⩽ 𝑓 ( 𝑥 ) holds if and only if 𝐵 lies below Conv( 𝐴 ) . When applied with 𝑈 = 𝑟 ⋅ 𝐴 and 𝑉 = 𝐵 , this lemmaimplies that the inequality 𝑟 ⋅ 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) holds if and only if 𝑟 ⋅ 𝐴 lies below Conv( 𝐵 ) . (cid:3) We say that a set
𝑈 ⊆ ℝ 𝑛 lies tightly above the convex hull Conv( 𝑉 ) of a set 𝑉 ⊆ ℝ 𝑛 if for everyvector 𝑢 ∈ 𝑈 , the inequality 𝑢 ⩾ 𝑐 holds for some convex combination 𝑐 of vectors 𝑣 ∈ 𝑉 with the same support as that of 𝑢 . That is, we now additionally have that none of the vectors 𝑣 ∈ 𝑉 in theconvex combination 𝑐 has a zero in a position 𝑖 where 𝑢 𝑖 ≠ . Lemma 6.5 (Minimization) . Let
𝐴 ⊂ ℕ 𝑛 be some finite set of vectors, Φ be a (min , +) circuit, and 𝐵 ⊂ ℕ 𝑛 the set of vectors produced by Φ .Then the following two assertions are equivalent. (1) Φ approximates the minimization problem on 𝐴 within a factor 𝑟 . (2) 𝐵 lies above Conv( 𝐴 ) and 𝑟 ⋅ 𝐴 lies above Conv( 𝐵 ) .If 𝐴 ⊆ {0 , 𝑛 and 𝐴 is an antichain, then these assertions are equivalent to: 𝐵 lies above 𝐴 and 𝑟 ⋅ 𝐴 lies tightly above Conv( 𝐵 ) .Proof. By Lemma 3.2, we can assume the circuit Φ is constant-free. Hence, by Proposition 3.1, thecircuit solves the minimization problem of the form Φ( 𝑥 ) = min 𝑏 ∈ 𝐵 ⟨ 𝑏, 𝑥 ⟩ . The minimization problemon 𝐴 is of the form 𝑓 ( 𝑥 ) = min 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ . The circuit approximates the minimization problem on 𝐴 within factor 𝑟 if and only if 𝑓 ( 𝑥 ) ⩽ Φ( 𝑥 ) ⩽ 𝑟 ⋅ 𝑓 ( 𝑥 ) holds for all weightings 𝑥 ∈ ℝ 𝑛 + . When appliedwith 𝑈 = 𝐵 and 𝑉 = 𝐴 , Lemma 6.3(1) implies that the inequality Φ( 𝑥 ) ⩾ 𝑓 ( 𝑥 ) holds if and only if 𝐵 lies above Conv( 𝐴 ) . When applied with 𝑈 = 𝑟 ⋅ 𝐴 and 𝑉 = 𝐵 , this lemma implies that the inequality 𝑟 ⋅ 𝑓 ( 𝑥 ) ⩾ Φ( 𝑥 ) holds if and only if 𝑟 ⋅ 𝐴 lies above Conv( 𝐵 ) . This shows the equivalence of theassertions (1) and (2).Suppose now that 𝐴 ⊆ {0 , 𝑛 , and that 𝐴 is an antichain. The implication (3) ⇒ (2) is obvious. So,assume that the set 𝐵 has property (2), i.e., that 𝐵 lies above Conv( 𝐴 ) and 𝑟 ⋅ 𝐴 lies above Conv( 𝐵 ) .To show that 𝐵 lies above the set 𝐴 (not only above its convex hull), take an arbitrary vector 𝑏 ∈ 𝐵 .Since 𝐵 lies above Conv( 𝐴 ) , there must be a vector 𝑎 ∈ 𝐴 and a scalar < 𝜆 ⩽ such that 𝑏 ⩾ 𝜆 ⋅ 𝑎 .Since 𝑎 is a - vector, and 𝑏 is a nonnegative integer vector, 𝑏 ⩾ 𝑎 must hold.To show that 𝑟 ⋅ 𝐴 lies tightly above Conv( 𝐵 ) , take an arbitrary vector 𝑎 ∈ 𝐴 . Since, by (2),the set 𝑟 ⋅ 𝐴 lies above Conv( 𝐵 ) , the inequality 𝑟 ⋅ 𝑎 ⩾ 𝑐 must hold for some convex combination 𝑐 = 𝜆 ⋅ ⃗𝑏 + ⋯ + 𝜆 𝑚 ⋅ ⃗𝑏 𝑚 of vectors in 𝐵 , where all scalars 𝜆 𝑖 are positive. It remains to show that then 𝑆 ⃗𝑏 𝑖 = 𝑆 𝑎 holds for all 𝑖 = 1 , … , 𝑚 , i.e., that all the vectors in this convex combination have the samesupport as our vector 𝑎 .Since (as we have just shown) the set 𝐵 lies above the set 𝐴 , there must be (not necessarily distinct)vectors ⃗𝑎 , … , ⃗𝑎 𝑚 in 𝐴 such that ⃗𝑏 𝑖 ⩾ ⃗𝑎 𝑖 for all 𝑖 = 1 , … , 𝑚 and, hence, 𝑐 = ∑ 𝑚𝑖 =1 𝜆 𝑖 ⋅ ⃗𝑏 𝑖 ⩾ ∑ 𝑚𝑖 =1 𝜆 𝑖 ⋅ ⃗𝑎 𝑖 .The inequality 𝑟 ⋅ 𝑎 ⩾ 𝑐 implies that 𝑆 𝑎 ⊇ 𝑆 ⃗𝑏 𝑖 ⊇ 𝑆 ⃗𝑎 𝑖 must hold for all 𝑖 . Since 𝐴 is an antichain andconsists of only - vectors, this implies ⃗𝑎 𝑖 = 𝑎 for all 𝑖 . We thus have 𝑆 ⃗𝑏 𝑖 = 𝑆 𝑎 for all 𝑖 = 1 , … , 𝑚 , asdesired. (cid:3) Remark . Lemma 6.5 implies that in order to show that theminimization problem on an antichain F ⊆ [ 𝑛 ] can be 𝑟 -approximated by a (min , +) circuit of size 𝑡 , it is enough to design a monotone arithmetic (+ , ×) circuit Φ of size ⩽ 𝑡 such that the polynomialcomputed by this circuit has the following two properties:(1) for every monomial ∏ 𝑖 ∈ 𝑇 𝑥 𝑑 𝑖 𝑖 there is a set 𝑆 ∈ F with 𝑆 ⊆ 𝑇 ;(2) for every set 𝑆 ∈ F there is a monomial ∏ 𝑖 ∈ 𝑇 𝑥 𝑑 𝑖 𝑖 with 𝑇 = 𝑆 and all 𝑑 𝑖 ⩽ 𝑟 .Indeed, property (1) ensures that the set 𝐵 of vectors produced by the arithmetic circuit Φ lies abovethe set 𝐴 of characteristic - vectors of sets in F , while property (2) ensures that the set 𝑟 ⋅ 𝐴 liestightly above Conv( 𝐵 ) . By Lemma 6.5, the (min , +) version of Φ 𝑟 -approximates the minimizationproblem on F . 7. A TIGHT BOOLEAN BOUND FOR (min , +) CIRCUITS
In Section 4.1, we have shown (Theorem 4.3) that the monotone boolean circuit complexity of thedecision versions of minimization problems is a lower bound on the size of (min , +) circuits approxi-mating these problems within any finite approximation factor 𝑟 ⩾ . Lemma 6.5 will allow us to takethe factor 𝑟 into account, that is, to show that approximating (min , +) circuits and monotone booleancircuits are even more tightly related (as given in Theorem 7.1 below). For this purpose, we introducethe concept of “semantic degree” of monotone boolean circuits. Semantic degree of boolean circuits. A minterm of a monotone boolean function 𝑓 ( 𝑥 , … , 𝑥 𝑛 ) is a vector 𝑎 ∈ {0 , 𝑛 such that 𝑓 ( 𝑎 ) = 1 , but 𝑓 ( 𝑎 ′ ) = 0 for any vector 𝑎 ′ obtained by switching anysingle -entry of 𝑎 to . The boolean function defined by a finite set 𝐴 ⊂ ℕ 𝑛 of vectors is of the form 𝑓 𝐴 ( 𝑥 ) = ⋁ 𝑎 ∈ 𝐴 ⋀ 𝑖 ∈ 𝑆 𝑎 𝑥 𝑖 , where, as before, 𝑆 𝑎 = { 𝑖 ∶ 𝑎 𝑖 ≠ is the support of vector 𝑎 . In particular, if 𝐴 ⊂ {0 , 𝑛 is the setof minterms of a boolean function, then this function is of the form 𝑓 𝐴 (is defined by the set of itsminterms).A monotone boolean (∨ , ∧) circuit 𝜙 for a boolean function 𝑓 𝐴 (defined by its set 𝐴 of minterms)not only computes the function 𝑓 𝐴 , but also produces (purely syntactically) some finite set 𝐵 ⊂ ℕ 𝑛 ofvectors, as given in Section 3.2. By Proposition 3.1, the circuit 𝜙 computes the boolean function 𝑓 𝐵 defined by the set 𝐵 . Since the circuit 𝜙 computes the function 𝑓 𝐴 , we know that 𝑓 𝐵 ( 𝑥 ) = 𝑓 𝐴 ( 𝑥 ) musthold for all 𝑥 ∈ {0 , 𝑛 . The “semantic degree” of the circuit 𝜙 (motivated by Lemma 6.5) gives anupper bound on the magnitudes of entries of particular (not all) vectors of the set 𝐵 .Namely, we define the semantic degree , deg( 𝜙 ) , of 𝜙 as the minimum real number 𝑟 such that the set 𝑟 ⋅ 𝐴 lies tightly above the convex hull Conv( 𝐵 ) of the set 𝐵 produced by the circuit 𝜙 . Recall that thismeans that for every minterm 𝑎 ∈ 𝐴 there are vectors ⃗𝑏 , … , ⃗𝑏 𝑚 ∈ 𝐵 and positive scalars 𝜆 , … , 𝜆 𝑚 such that 𝜆 + ⋯ + 𝜆 𝑚 = 1 , 𝑆 ⃗𝑏 = … = 𝑆 ⃗𝑏 𝑚 = 𝑆 𝑎 and(8) 𝑎 ⩽ 𝜆 ⋅ ⃗𝑏 + ⋯ + 𝜆 𝑚 ⋅ ⃗𝑏 𝑚 ⩽ 𝑟 ⋅ 𝑎 . We use the adjective “semantic” because deg( 𝜙 ) depends on the function computed by 𝜙 , that is, onthe set 𝐴 of minterms of this function. Note that the first inequality in Eq. (8) always holds because 𝑎 is a - vector, and 𝑆 ⃗𝑏 𝑖 = 𝑆 𝑎 holds for all vectors 𝑏 𝑖 (we included this inequality just for clarity of theconcept).7.2. The converse of the boolean bound (Theorem 4.3).
For a finite set
𝐴 ⊂ ℕ 𝑛 , let 𝖡𝗈𝗈𝗅 𝑟 ( 𝐴 ) denotethe minimum size of a monotone boolean circuit of semantic degree at most 𝑟 computing the booleanfunction 𝑓 𝐴 defined by 𝐴 . Theorem 7.1 (Tight boolean bound) . If 𝐴 ⊂ {0 , 𝑛 is an antichain, then 𝖬𝗂𝗇 𝑟 ( 𝐴 ) = 𝖡𝗈𝗈𝗅 𝑟 ( 𝐴 ) holdsfor every 𝑟 ⩾ .Proof. To show
𝖬𝗂𝗇 𝑟 ( 𝐴 ) ⩽ 𝖡𝗈𝗈𝗅 𝑟 ( 𝐴 ) , take a monotone boolean (∨ , ∧) circuit 𝜙 of semantic degree 𝑟 computing the boolean function 𝑓 𝐴 defined by 𝐴 . We can assume that the circuit is constant-free: and are the only possible constants, and they can be trivially eliminated from the circuit. Let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by the circuit 𝜙 . By Proposition 3.1, the circuit 𝜙 computes the booleanfunction 𝑓 𝐵 defined by this set, that is, 𝑓 𝐵 ( 𝑥 ) = 𝑓 𝐴 ( 𝑥 ) holds for all inputs 𝑥 ∈ {0 , 𝑛 .Let Φ be the (also constant-free) tropical (min , +) version of the boolean circuit 𝜙 obtained from 𝜙 by replacing each ∨ -gate by a min -gate, and each ∧ -gate by a + -gate. The (min , +) circuit Φ producesthe same set 𝐵 .Since 𝑓 𝐵 ( 𝑥 ) ⩽ 𝑓 𝐴 ( 𝑥 ) must hold for all inputs 𝑥 ∈ {0 , 𝑛 , for every 𝑏 ∈ 𝐵 there must be a vector 𝑎 ∈ 𝐴 with 𝑆 𝑏 ⊇ 𝑆 𝑎 . Since vectors in 𝐴 are - vectors, this latter inclusion yields 𝑏 ⩾ 𝑎 . Thus, theset 𝐵 lies above 𝐴 and, hence, also above Conv( 𝐴 ) . Since the circuit 𝜙 has semantic degree 𝑟 , weadditionally have that the set 𝑟 ⋅ 𝐴 lies above Conv( 𝐵 ) . By Lemma 6.5, the circuit Φ approximates theminimization problem on 𝐴 within the factor 𝑟 .To show 𝖡𝗈𝗈𝗅 𝑟 ( 𝐴 ) ⩽ 𝖬𝗂𝗇 𝑟 ( 𝐴 ) , take a tropical (min , +) circuit Φ approximating the minimizationproblem on 𝐴 within the factor 𝑟 , and let 𝐵 ⊂ ℕ 𝑛 be the set of vectors produced by Φ . By Lemma 3.2, we can assume that the circuit Φ is constant-free. Let 𝜙 be the boolean version of the tropical circuit Φ obtained by replacing each min -gate by an ∨ -gate, and each + -gate by an ∧ -gate. (Recall that min and ∨ are “additions” and + and ∧ are “multiplications” in the corresponding semirings.) The circuit 𝜙 produces the same set 𝐵 . By Lemma 6.5, we know that the set 𝐵 has the following two properties:(i) 𝐵 lies above 𝐴 ;(ii) 𝑟 ⋅ 𝐴 lies tightly above Conv( 𝐵 ) .By property (ii), the semantic degree of the boolean circuit 𝜙 is at most 𝑟 . On the other hand, property(i) implies that the support of every vector 𝑏 ∈ 𝐵 contains the support of at least one vector 𝑎 ∈ 𝐴 ,and property (ii) implies that the support of every vector 𝑎 ∈ 𝐴 contains the support of at least onevector 𝑏 ∈ 𝐵 . In terms of Proposition 4.1, this means that the set 𝐵 is similar to the set 𝐴 . Since (byProposition 3.1) the circuit computes a boolean function 𝑓 𝐵 defined by the set 𝐵 , Proposition 4.1 itselfimplies that 𝜙 computes the boolean function defined by the set 𝐴 . (cid:3) Bounds on semantic degree. An 𝑟 - bounded copy of a boolean vector 𝑎 ∈ {0 , 𝑛 is an integervector 𝑏 ∈ ℕ 𝑛 which has the same nonzero positions as 𝑎 , and every nonzero position of 𝑏 is at most 𝑟 .In particular, the unique -bounded copy of 𝑎 is the vector 𝑎 itself. Recall that | 𝑎 | = ⟨ 𝑎, 𝑎 ⟩ is thenumber of ones in a - vector 𝑎 . Proposition 7.2.
Let 𝜙 be a monotone boolean circuit computing a boolean function 𝑓 , 𝐴 ⊂ {0 , 𝑛 the set of minterms of 𝑓 , and 𝐵 ⊂ ℕ 𝑛 the set of vectors produced by the circuit 𝜙 . Then the followingholds. (1) deg( 𝜙 ) = 1 if and only if 𝐴 ⊆ 𝐵 . (2) deg( 𝜙 ) ⩽ 𝑟 holds if for every 𝑎 ∈ 𝐴 the set 𝐵 contains at least one 𝑟 -bounded copy of 𝑎 . (3) If deg( 𝜙 ) ⩽ 𝑟 holds, then for every 𝑎 ∈ 𝐴 the set 𝐵 contains at least one 𝑠 -bounded copy of 𝑎 for 𝑠 ⩽ 𝑟 | 𝑎 | − | 𝑎 | + 𝑟 . Note that (1) is a special case of (2) and (3) for 𝑟 = 1 Proof.
The “if” direction in (1) is obvious. The “only if” direction follows from a simple observation:a convex combination 𝜆 𝑏 + ⋯ + 𝜆 𝑚 𝑏 𝑚 of positive integers 𝑏 𝑖 with all 𝜆 𝑖 > can be equal only if 𝑏 = … = 𝑏 𝑚 = 1 .Claim (2) is also obvious, because for every - vector 𝑎 , the inequality 𝑏 ⩽ 𝑟 ⋅ 𝑎 holds for every 𝑟 -bounded copy 𝑏 of 𝑎 .To show claim (3), assume that deg( 𝜙 ) ⩽ 𝑟 . Take any minterm 𝑎 ∈ 𝐴 and let 𝑚 = | 𝑎 | be thenumber of ones in 𝑎 . By the definition of the semantic degree, we know that there must be a convexcombination 𝑐 = ∑ 𝑙𝑖 =1 𝜆 𝑖 ⋅ ⃗𝑏 𝑖 of vectors 𝑏 𝑖 ∈ 𝐵 with all supports 𝑆 ⃗𝑏 𝑖 = 𝑆 𝑎 such that 𝑐 ⩽ 𝑟 ⋅ 𝑎 holds.By Carathéodory’s theorem [5], if a vector is in the convex hull of some set 𝑃 ⊆ ℝ 𝑚 of vectors, thenthis vector can be written as a convex combination of 𝑚 + 1 or fewer vectors in 𝑃 . So, by taking 𝑃 = { ⃗𝑏 , … , ⃗𝑏 𝑙 } , we can assume that 𝑙 ⩽ | 𝑎 | + 1 = 𝑚 + 1 .Consider the vectors ⃗𝑏 ′ 𝑖 ∶= ⃗𝑏 𝑖 − 𝑎 ⩾ ⃗ (the vectors ⃗𝑏 ′ 𝑖 are nonnegative, because vectors ⃗𝑏 𝑖 have thesame support as 𝑎 ). Then 𝑐 = 𝑎 + 𝑐 ′ with 𝑐 ′ ∶= ∑ 𝑙𝑖 =1 𝜆 𝑖 ⃗𝑏 ′ 𝑖 = 𝑐 − 𝑎 ⩽ 𝑟 ⋅ 𝑎 − 𝑎 = ( 𝑟 − 1) ⋅ 𝑎 . Since 𝜆 + ⋯ + 𝜆 𝑙 = 1 , there must be an 𝑖 such that 𝜆 𝑖 ⩾ 𝑙 ⩾ 𝑚 + 1) . From 𝜆 𝑖 ⋅ ⃗𝑏 ′ 𝑖 ⩽ 𝑐 ′ ⩽ ( 𝑟 − 1) ⋅ 𝑎 , andsince 𝑎 is a - vector, we have that all entries of vector ⃗𝑏 ′ 𝑖 must be at most ( 𝑟 − 1)∕ 𝜆 𝑖 ⩽ ( 𝑟 − 1)( 𝑚 + 1) .Hence, all entries of the vector ⃗𝑏 𝑖 are at most ( 𝑟 − 1)( 𝑚 + 1) + 1 = 𝑟𝑚 − 𝑚 + 𝑟 , as desired. (cid:3) Remark . An apparent advantage of Proposition 7.2 is that it avoids the somewhat involved definitionof the semantic degree via convex hulls. Items (i) and (ii) may be useful when proving upper bounds, while items (i) and (iii) may be useful when proving lower bounds on the size of monotone booleancircuits of bounded semantic degree. Remark . Note that the upper bound 𝑠 ⩽ 𝑟 | 𝑎 | − | 𝑎 | + 𝑟 in item (iii) of Proposition 7.2 cannot besubstantially improved. Take 𝑚 = | 𝑎 | vectors ⃗𝑏 𝑖 ∶= 𝑎 + 𝑚 ( 𝑟 − 1) ⃗𝑒 𝑖 , and let all 𝜆 𝑖 ∶= 1∕ 𝑚 . Then theconvex combination 𝑐 = ∑ 𝑚𝑖 =1 𝜆 𝑖 ⋅ ⃗𝑏 𝑖 = 𝑎 + ( 𝑟 − 1) ⋅ 𝑎 = 𝑟 ⋅ 𝑎 satisfies 𝑐 ⩽ 𝑟 ⋅ 𝑎 , but every vector ⃗𝑏 𝑖 inthis combination has 𝑠 = 1 + 𝑚 ( 𝑟 − 1) = 𝑟 | 𝑎 | − | 𝑎 | + 1 as one of it entries.The following example shows that the semantic degree of monotone boolean circuits can be smalleven when some vectors produced by the circuit have very large entries. Example . Let 𝐴 be the set of characteristic - vectors of all simple paths in 𝐾 𝑛 between two fixed vertices 𝑠 and 𝑡 . Then the boolean function 𝑓 𝐴 defined by 𝐴 is the 𝑠 - 𝑡 connectivityfunction STCONN on 𝑛 -vertex graphs. The Bellman–Ford pure DP algorithm for the shortest 𝑠 - 𝑡 pathproblem gives us a monotone boolean (∨ , ∧) circuit 𝜙 of size 𝑂 ( 𝑛 ) computing the boolean function 𝑓 𝐴 . The circuit has gates 𝑢 𝑙𝑗 at which the existence of a path from vertex 𝑠 to vertex 𝑗 with at most 𝑙 edges is detected. Then 𝑢 𝑗 = 𝑥 𝑠,𝑗 for all 𝑗 ≠ 𝑠 , and the recursion of Bellman–Ford is to compute 𝑢 𝑙 +1 𝑗 as the OR of 𝑢 𝑙𝑗 and all 𝑢 𝑙𝑖 ∧ 𝑥 𝑖,𝑗 for 𝑖 ∉ { 𝑠, 𝑗 } . The output gate is 𝑢 𝑛 −1 𝑡 .The vectors of the set 𝐵 ⊂ ℕ 𝑛 produced by the Bellman–Ford circuit 𝜙 correspond not to (simple)paths but rather to walks of length at most 𝑛 − 1 from 𝑠 to 𝑡 . Since a walk can traverse the same edgemany times, some vectors in 𝐵 have entries much larger than . Still, by Proposition 7.2(1), deg( 𝜙 ) = 1 holds: every (simple) 𝑠 - 𝑡 path is also a walk of length at most 𝑛 − 1 , implying that 𝐴 ⊆ 𝐵 .7.4.
Semantic versus syntactic degree.
The standard, “syntactic” definition of the degree is the fol-lowing. Each input node holding a variable has degree . The degree of an OR gate is the maximum of the degrees of its input gates, and the degree of an AND gate is the sum of the degrees of its inputgates. The following proposition shows that the semantic degree never exceeds the syntactic degree. Proposition 7.3.
Let 𝜙 and 𝜙 be any two monotone boolean circuits. Then deg( 𝜙 ∨ 𝜙 ) ⩽ max { deg( 𝜙 ) , deg( 𝜙 ) } and deg( 𝜙 ∧ 𝜙 ) ⩽ deg( 𝜙 ) + deg( 𝜙 ) .Proof. For 𝑖 ∈ {1 , , let 𝐴 𝑖 ⊆ {0 , 𝑛 be the set of minterms of the boolean function computed by 𝜙 𝑖 ,and let 𝐵 𝑖 ⊂ ℕ 𝑛 be the set of vectors produced by 𝜙 𝑖 . Let 𝑟 𝑖 = deg( 𝜙 𝑖 ) be the semantic degree of 𝜙 𝑖 .Take an arbitrary minterm 𝑎 of 𝜙 . If 𝜙 = 𝜙 ∨ 𝜙 , then 𝐵 = 𝐵 ∪ 𝐵 is the union of the set 𝐵 and 𝐵 , and 𝑎 ∈ 𝐴 𝑖 for some 𝑖 ∈ {1 , . We know that 𝑐 ⩽ 𝑟 𝑖 ⋅ 𝑎 must hold for some vector 𝑐 in Conv( 𝐵 𝑖 ) ⊆ Conv( 𝐵 ) . So, deg( 𝜙 ∨ 𝜙 ) ⩽ max{ 𝑟 , 𝑟 } in this case.If 𝜙 = 𝜙 ∧ 𝜙 , then 𝐵 = 𝐵 + 𝐵 is the Minkowski sum of the sets 𝐵 and 𝐵 , and 𝑎 = ⃗𝑎 ∨ ⃗𝑎 is a componentwise OR of some minterms ⃗𝑎 ∈ 𝐴 and ⃗𝑎 ∈ 𝐴 . We know that ⃗𝑐 ⩽ 𝑟 ⋅ ⃗𝑎 and ⃗𝑐 ⩽ 𝑟 𝑖 ⋅ ⃗𝑎 must hold for some vectors ⃗𝑐 ∈ Conv( 𝐵 ) and ⃗𝑐 ∈ Conv( 𝐵 ) . A well-known propertyof Minkowski sums is that Conv( 𝐵 ) + Conv( 𝐵 ) = Conv( 𝐵 + 𝐵 ) always holds. Hence, the vector 𝑐 = ⃗𝑐 + ⃗𝑐 belongs to Conv( 𝐵 ) and satisfies 𝑐 = ⃗𝑐 + ⃗𝑐 ⩽ 𝑟 ⋅ ⃗𝑎 + 𝑟 ⋅ ⃗𝑎 ⩽ 𝑟 ⋅ 𝑎 + 𝑟 ⋅ 𝑎 = ( 𝑟 + 𝑟 ) ⋅ 𝑎 .So, deg( 𝜙 ∧ 𝜙 ) ⩽ 𝑟 + 𝑟 holds in this case. (cid:3) The following example illustrates that, together with Proposition 7.3, the upper bound
𝖬𝗂𝗇 𝑟 ( 𝐴 ) ⩽ 𝖡𝗈𝗈𝗅 𝑟 ( 𝐴 ) given by Theorem 7.1 allows one to show that some minimization problems can be approxi-mated by small (min , +) circuits within (large) but finite factors by proving upper bounds for monotone boolean circuits of bounded semantic degree. Recall that some minimization problems cannot be ap-proximated by (min , +) circuits of polynomial size within any finite factor 𝑟 = 𝑟 ( 𝑛 ) at all (Section 4.2). Example . In the minimum weight spanning tree problem T 𝑛 , we are given an assign-ment of nonnegative real weights to the edges of 𝐾 𝑛 , and the goal is to compute the minimum weightof a spanning tree in 𝐾 𝑛 ; the weight of a subgraph is the sum of weights of its edges. We have shownin [17] that 𝖬𝗂𝗇 ( T 𝑛 ) = 2 Ω( √ 𝑛 ) .On the other hand, the decision version of this problem is the graph connectivity problem. Using the(pure) DP algorithm of Bellman and Ford, for every pair ( 𝑠, 𝑡 ) of vertices, the 𝑠 - 𝑡 connectivity problemcan be solved by a monotone boolean circuit 𝜙 𝑠,𝑡 of size 𝑂 ( 𝑛 ) and semantic degree deg( 𝜙 𝑠,𝑡 ) = 1 (see Example 1). So, the connectivity problem can be solved by the circuit 𝜙 , ∧ 𝜙 , ∧ ⋯ ∧ 𝜙 ,𝑛 ofsize 𝑂 ( 𝑛 ) . By Proposition 7.3, the circuit has semantic degree 𝑟 ⩽ 𝑛 − 1 . Theorem 7.1 implies that 𝖬𝗂𝗇 𝑟 ( T 𝑛 ) = 𝑂 ( 𝑛 ) holds for some finite factor 𝑟 ⩽ 𝑛 − 1 .8. C ONCLUSION AND OPEN PROBLEMS
Developing a workable taxonomy of existing algorithmic paradigms in rigorous mathematical termsis an important long-term goal. When pursuing this goal, the main difficulty is to prove unconditional lower bounds on the complexity of algorithms from particular classes, that is, to prove lower boundsnot relying on unproven complexity assumptions like P ≠ NP .In this paper, we consider the class of all pure DP algorithms, take tropical circuits as their natu-ral mathematical model, and prove the first non-trivial (even super-polynomial) unconditional lowerbounds for approximating pure DP algorithms in this model. The results imply that the approximationpowers of greedy and pure DP algorithms are incomparable. Some interesting questions still remainopen.Given a family F ⊆ [ 𝑛 ] of feasible solutions, and an approximation factor 𝑟 ⩾ let, as before, 𝖬𝖺𝗑 𝑟 ( F ) denote the minimum number of gates in a (max , +) circuit approximating the maximizationproblem 𝑓 ( 𝑥 ) = max 𝑆 ∈ F ∑ 𝑖 ∈ 𝑆 𝑥 𝑖 on F within the factor 𝑟 . In the case of minimization problems and (min , +) circuits, the corresponding complexity measure is 𝖬𝗂𝗇 𝑟 ( F ) .8.1. Minimization.
We have shown in Theorem 4.6 that there exist a lot of monotone boolean func-tions 𝑓 such that minterms of 𝑓 are bases of a matroid, and 𝑓 requires monotone boolean circuits ofexponential size. But we do not know of any explicit matroid for which the corresponding booleanfunction requires large monotone boolean circuits. Problem 1.
Prove a super-polynomial lower bound on the monotone boolean circuit complexity of an explicit boolean function whose minterms are bases of a matroid.
Let T 𝑛 be the family of all spanning trees in a complete 𝑛 -vertex graph 𝐾 𝑛 . Since T 𝑛 is a matroid,both minimization and maximization problems can be solved exactly (within factor 𝑟 = 1 ) by thegreedy algorithm. On the other hand, we know that 𝖬𝗂𝗇 ( T 𝑛 ) = 2 Ω( √ 𝑛 ) [17]. We also know that 𝖬𝗂𝗇 𝑟 ( T 𝑛 ) = 𝑂 ( 𝑛 ) holds if factor 𝑟 = 𝑛 − 1 is allowed (Example 2). Problem 2. Is 𝖬𝗂𝗇 ( T 𝑛 ) polynomial in 𝑛 ? Maximization.
The next question concerns the maximization problem on the matroid T 𝑛 of span-ning trees in 𝐾 𝑛 . We know that, for factor 𝑟 = 1 , we have 𝖬𝖺𝗑 ( T 𝑛 ) = 2 Ω( √ 𝑛 ) [17]. Problem 3. Is 𝖬𝖺𝗑 ( T 𝑛 ) polynomial in 𝑛 ? In Theorem 5.13, we considered the maximum weight problem on 𝑘 -partite hypergraphs . For 𝑘 = 2 ,the calculations made in the proof of Theorem 5.13 result in a trivial bound. This rises a naturalquestion: does a similar lower bound hold also for matchings in bipartite graphs? Let M 𝑛 be the family of all perfect matchings in a complete bipartite 𝑛 × 𝑛 graph. The greedy algorithm can approximatethe maximization problem on M 𝑛 within the factor . Problem 4. Is 𝖬𝖺𝗑 ( M 𝑛 ) polynomial in 𝑛 ? We have shown in Theorem 4.6 that the minimization problem on many matroids cannot be effi-ciently approximated by pure DP algorithms within any finite factor 𝑟 . But what happens with maxi-mization problems? By Proposition 5.2, we know that there are a lot of matroids F ⊆ [ 𝑛 ] such that 𝖬𝖺𝗑 ( F ) = 2 Ω( 𝑛 ) but 𝖬𝖺𝗑 𝑟 ( F ) ⩽ 𝑛 holds already for 𝑟 = 1 + 𝑜 (1) . Problem 5.
Are there matroids, on which the maximization problem cannot be efficiently approximatedby pure DP algorithms within some factor 𝑟 ⩾ 𝜖 for a constant 𝜖 > ? Note that here we only ask for the mere existence . By Proposition 5.2, the answer is “yes” for 𝑟 = 1 .But this proposition and Proposition 5.3 indicate that direct counting arguments may fail to answer thisquestion for slightly larger approximation factors 𝑟 .8.3. Tradeoffs between minimization and maximization.
If a family F of feasible solutions is uni-form (all sets of F have the same cardinality), then 𝖬𝗂𝗇 ( F ) = 𝖬𝖺𝗑 ( F ) (see, for example, [16,Lemma 2]). That is, if we consider exactly solving tropical circuits (factor 𝑟 = 1 ), then there is nodifference between the tropical circuit complexity of the minimization and the maximization problemon the same (uniform) set F of feasible solutions.But the situation is entirely different if we consider approximating circuits: Theorem 4.6 and Propo-sition 5.2 give us doubly-exponentially many in 𝑛 matroids F ⊆ [ 𝑛 ] such that 𝖬𝖺𝗑 𝑜 (1) ( F ) ⩽ 𝑛 , but 𝖬𝗂𝗇 𝑟 ( F ) = 2 Ω( 𝑛 ) for any finite factor 𝑟 = 𝑟 ( 𝑛 ) ⩾ . Problem 6.
Are there uniform families F for which the gap 𝖬𝖺𝗑 𝑟 ( F )∕ 𝖬𝗂𝗇 𝑠 ( F ) is exponential for 𝑟 ⩾ 𝑠 > ? Note that the separating family F is here required to be uniform (or at least to form an antichain):without this requirement, the gap can be artificially made large. To see this, take an arbitrary uniformfamily H ⊆ [ 𝑛 ] with large 𝖬𝖺𝗑 𝑟 ( H ) (as in Theorems 5.12 and 5.13), and extend it to a nonuniformfamily F by adding all single element sets. Then 𝖬𝗂𝗇 ( F ) ⩽ 𝑛 (just compute the minimum weight ofa single element), but 𝖬𝖺𝗑 𝑟 ( F ) still remains large.8.4. Pure DP algorithms with subtraction.
Can the size of tropical approximating circuits be sub-stantially reduced by allowing (besides min ∕ max and + ) also subtraction (−) gates? In the caseof the approximation factor 𝑟 = 1 (exact solution), we already know the answer: subtraction gates can then even exponentially decrease the circuit size. Namely, we already know that both directedand undirected versions of the MST problem (minimum weight spanning tree problem) on 𝑛 -vertexgraphs require tropical (min , +) circuits of size Ω( √ 𝑛 ) [15, 17] but, as shown by Fomin, Grigoriev andKoshevoy [10], both these problems are solvable by tropical (min , + , −) circuits of size only 𝑂 ( 𝑛 ) .Unfortunately, no non-trivial lower bounds for (min , + , −) circuits are known so far. So, at least twonatural questions arise. ◦ Prove lower bounds for (min , + , −) circuits, at least when 𝑟 = 1 . ◦ What about larger approximation factors 𝑟 > ?Note that, when restricted to the boolean domain {0 , , (min , + , −) circuits have the entire power of unrestricted boolean (∨ , ∧ , ¬) circuits: 𝑥 ∧ 𝑦 = min( 𝑥, 𝑦 ) , 𝑥 ∨ 𝑦 = min(1 , 𝑥 + 𝑦 ) and ¬ 𝑥 = 1 − 𝑥 . Thepoint, however, is that (min , + , −) circuits must correctly work over the entire real domain ℝ + . A PPENDIX
A. G
REEDY ALGORITHMS
Since we compared the approximation power of tropical circuits (and pure DP algorithms) with thatof the greedy algorithm, here we specify what we actually mean by “the” greedy algorithm.Let F ⊆ 𝐸 be some family of feasible solutions forming an antichain (no two members of F arecomparable under set inclusion). Given an ordering 𝑒 , … , 𝑒 𝑛 of the elements of 𝐸 , there are twotrivial heuristics to end up with a member of F by treating the elements one-by-one in this fixed order. First-in : Start with the empty partial solution, treat the elements one-by-one and, at each step, add the next element to the current partial solution if and only if the extended partial solutionstill lies in at least one feasible solution.
First-out : Start with the entire set 𝐸 as a partial solution, treat the elements one-by-one and, ateach step, remove the next element from the current partial solution if and only if the reducedpartial solution still contains at least one feasible solution.Recall that an optimization (maximization or minimization) problem on F is, given an assignmentof nonnegative real weights to the ground elements, to compute the maximum or the minimum weightof a feasible solution, the latter being the sum of weights of its elements.In this paper, by the greedy algorithm we always mean the algorithm which, on every input weight-ing 𝑥 ∶ 𝐸 → ℝ + , starts with the heaviest-first ordering 𝑥 ( 𝑒 ) ⩾ 𝑥 ( 𝑒 ) ⩾ … ⩾ 𝑥 ( 𝑒 𝑛 ) of the elements of 𝐸 , and uses:- the first-in heuristic (“best-in” strategy) in the case of maximization ;- the first-out heuristic (“worst-out” strategy) in the case of minimization .That is, at each step, the “oracle” of the maximizing greedy algorithm decides whether the currentset is still contained in at least one feasible solution, while that of the minimizing greedy algorithmdecides whether the current set still contains at least one feasible solution.We denote the approximation factor achieved by the greedy algorithm on a corresponding optimiza-tion (minimization or maximization) problem on F by 𝑟 greed ( F ) . It is well known (see, for example,[23, Theorem 1.8.4] that 𝑟 greed ( F ) = 1 if and only if F is (the family of bases of) a matroid. If F is nota matroid, then greedy algorithms can only approximate the corresponding optimization problems. Inthis case, it is already crucial what greedy strategy is used. Example . The choice of these special heuristics (first-in for maximization and first-out for min-imization) is not an accident. Namely, a greedy algorithm starting with the lightest-first ordering 𝑥 ( 𝑒 ) ⩽ 𝑥 ( 𝑒 ) ⩽ … ⩽ 𝑥 ( 𝑒 𝑛 ) , and using the first-out heuristic (“worst-out” strategy) for maximizationor first-in heuristic (“best-in” strategy) for minimization would be unable to approximate some opti-mization problems within any finite factor. To give a simple example, consider the path with threenodes ∙ ∙ ∙ 𝑎 𝑏 𝑐 , and let F be the family consisting of just two sets { 𝑎, 𝑐 } and { 𝑏 } (the maximal in-dependent sets in this path). If we take an arbitrarily large number 𝑀 > , and give weights 𝑥 ( 𝑎 ) = 0 , 𝑥 ( 𝑏 ) = 1 and 𝑥 ( 𝑐 ) = 𝑀 , then both these greedy algorithms will treat the vertices in the order 𝑎, 𝑏, 𝑐 .The worst-out maximizing greedy on F will output 𝑥 ( 𝑏 ) = 1 while the optimum is 𝑀 , and the best-ingreedy for minimization will output 𝑥 ( 𝑎 ) + 𝑥 ( 𝑐 ) = 0 + 𝑀 , while the optimum is . In both cases, theachieved approximation factor is 𝑟 ⩾ 𝑀 (unbounded).If, however, the greedy algorithm uses the “right” strategies for maximization and for minimization,then the approximation factor is always bounded (albeit possibly growing with the size of feasiblesolutions). Say that a family F of sets is 𝑚 - bounded if | 𝑆 | ⩽ 𝑚 holds for all 𝑆 ∈ F . Proposition A.1.
For every 𝑚 -bounded family F , we have 𝑟 greed ( F ) ⩽ 𝑚 , and there exist 𝑚 -boundedantichains F for which 𝑟 greed ( F ) = 𝑚 . Proof.
To show the upper bound, take an arbitrary weighting 𝑥 ∶ 𝐸 → ℝ + . Consider the heaviest-first ordering 𝑥 ( 𝑒 ) ⩾ … ⩾ 𝑥 ( 𝑒 𝑖 ) ⩾ … ⩾ 𝑥 ( 𝑒 𝑛 ) . Let 𝑒 𝑖 be the first element accepted by the greedyalgorithm. Let 𝑆 ∈ F be an optimal solution for the input 𝑥 , and 𝐴 ∈ F be the solution found by thealgorithm. Let also 𝑥 ( 𝑆 ) = ∑ 𝑖 ∈ 𝑆 𝑥 ( 𝑒 𝑖 ) and 𝑥 ( 𝐴 ) = ∑ 𝑖 ∈ 𝐴 𝑥 ( 𝑒 𝑖 ) be their weights.If this is the maximizing (best-in) greedy, then 𝑒 𝑖 is the first element belonging to at least one feasibleset. So, 𝑆 ∩ { 𝑒 , … , 𝑒 𝑖 −1 } = ∅ , implying that 𝑥 ( 𝑆 ) ⩽ | 𝑆 | ⋅ 𝑥 ( 𝑒 𝑖 ) ⩽ 𝑚 ⋅ 𝑥 ( 𝑒 𝑖 ) ⩽ 𝑚 ⋅ 𝑥 ( 𝐴 ) , as desired.If this is the minimizing (worst-out) greedy, then { 𝑒 𝑖 +1 , … , 𝑒 𝑛 } cannot contain any feasible solution(for otherwise, 𝑒 𝑖 would not be accepted). So, some element 𝑒 𝑗 with 𝑗 ⩽ 𝑖 must belong to the optimalsolution 𝑆 . But then 𝑥 ( 𝑆 ) ⩾ 𝑥 ( 𝑒 𝑗 ) ⩾ 𝑥 ( 𝑒 𝑖 ) , whereas 𝑥 ( 𝐴 ) ⩽ | 𝐴 | ⋅ 𝑥 ( 𝑒 𝑖 ) ⩽ 𝑚 ⋅ 𝑥 ( 𝑒 𝑖 ) , implying that 𝑥 ( 𝐴 ) ⩽ 𝑚 ⋅ 𝑥 ( 𝑆 ) , as desired.To show that 𝑟 greed ( F ) ⩾ 𝑚 holds for some 𝑚 -bounded antichains F , take an arbitrarily small number 𝜖 > , and consider the star 𝐾 ,𝑚 centered in 𝑎 and with leaves 𝑏 , … , 𝑏 𝑚 . Let F consist of the only twomaximal independent sets { 𝑎 } and { 𝑏 , … , 𝑏 𝑚 } in this graph. Give the weight 𝑐 ∶= 1∕(1 − 𝜖 ∕2) > to the center , and weights to the leaves. The maximizing (best-in) greedy will output 𝑐 while theoptimum is 𝑚 , and the minimizing (worst-out) greedy algorithms will output 𝑚 while the optimum is 𝑐 . In both cases, the achieved approximation factor is 𝑟 ⩾ 𝑚 ∕ 𝑐 > (1 − 𝜖 ) 𝑚 . (cid:3) A PPENDIX
B. S
IDON SETS : PROOF OF T HEOREM
𝐴 ⊂ ℕ 𝑛 of vectors is a Sidon set if for all vectors 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐴 : if 𝑎 + 𝑏 = 𝑐 + 𝑑 , then { 𝑎, 𝑏 } = { 𝑐, 𝑑 } . That is, knowing the sum of two vectors in 𝐴 , we know which vectors were added.Let (as before) 𝖬𝖺𝗑 𝑟 ( 𝐴 ) denote the minimum size of a tropical (max , +) circuit 𝑟 -approximating theproblem 𝑓 ( 𝑥 ) = max 𝑎 ∈ 𝐴 ⟨ 𝑎, 𝑥 ⟩ on 𝐴 .Let 𝑚 be an odd integer, and 𝑛 = 4 𝑚 . Our goal is to show that then there is an explicit Sidonset 𝐴 ⊆ {0 , 𝑛 such that 𝖬𝖺𝗑 ( 𝐴 ) ⩾ 𝑛 ∕4 but 𝖬𝖺𝗑 ( 𝐴 ) ⩽ 𝑛 . For this, consider the cubic parabola 𝐶 = {( 𝑧, 𝑧 ) ∶ 𝑧 ∈ {0 , 𝑚 } ⊆ GF(2 𝑚 ) . As customary, we view vectors in 𝑧 ∈ {0 , 𝑚 as coefficient-vectors of polynomials of degree at most 𝑚 −1 over GF(2) when rising them to a power. Note, however,that in the definition of Sidon sets, the sum of vectors is taken over the semigroup ( ℕ , +) , not over GF(2) ; in particular, 𝑎 + 𝑎 = 0 holds only for 𝑎 = 0 .For a finite set 𝐴 ⊂ ℕ 𝑛 of vectors, let 𝐿 ( 𝐴 ) denote the minimum size of a Minkowski (∪ , +) circuitproducing 𝐴 . We will use the following three facts. Recall that a set 𝐴 ⊆ {0 , 𝑛 is uniform if all itsvectors have the same number of ones.(1) The cubic parabola 𝐶 ⊆ {0 , 𝑚 is a Sidon set [21, Theorem 2].(2) 𝐿 ( 𝐴 ) ⩾ | 𝐴 | holds for every Sidon set 𝐴 ⊂ ℕ 𝑛 [12, Theorem 1].(3) If 𝐴 ⊆ {0 , 𝑛 is uniform, then 𝖬𝖺𝗑 ( 𝐴 ) ⩾ 𝐿 ( 𝐴 ) [15, Theorem 2.9].The cubic parabola 𝐶 is not uniform, and we cannot apply (3) to it. But, using a simple trick (suggestedby Igor Sergeev, personal communication), we can extend this set to a uniform Sidon set. For a - vector 𝑎 , let 𝑎 denote the componentwise negation of 𝑎 . For example, if 𝑎 = (0 , , then 𝑎 = (1 , , .Consider the following set of vectors: 𝐴 = {( 𝑐, 𝑐 ) ∶ 𝑐 ∈ 𝐶 } = { ( 𝑎, 𝑎 , 𝑎, 𝑎 ) ∶ 𝑎 ∈ {0 , 𝑚 } ⊆ {0 , 𝑛 . This set is already uniform: every vector of 𝐴 has exactly 𝑚 ones. The set 𝐴 is also a Sidon setbecause, by (1), the set 𝐶 was such. So, (2) and (3) imply that 𝖬𝖺𝗑 ( 𝐴 ) ⩾ | 𝐴 | = 2 𝑚 = 2 𝑛 ∕4 .It remains therefore to prove the upper bound 𝖬𝖺𝗑 ( 𝐴 ) ⩽ 𝑛 . We have 𝑛 = 4 𝑚 variables 𝑥 , … , 𝑥 𝑚 .Our approximating circuit will solve the maximization problem on the set 𝐵 = 𝐵 ′ ∪ 𝐵 ′′ , where 𝐵 ′ = {( 𝑎, , 𝑎,
0) ∶ 𝑎 ∈ {0 , 𝑚 } and 𝐵 ′′ = {(0 , 𝑎, , 𝑎 ) ∶ 𝑎 ∈ {0 , 𝑚 } . The maximization problem on 𝐵 is to compute 𝑓 ( 𝑥 ) = max{ 𝑔 ( 𝑥 ) , ℎ ( 𝑥 )} , where 𝑔 ( 𝑥 ) = max 𝑚 ∑ 𝑖 =1 𝑎 𝑖 𝑥 𝑖 + 𝑚 ∑ 𝑖 =2 𝑚 +1 (1 − 𝑎 𝑖 ) 𝑥 𝑖 ; ℎ ( 𝑥 ) = max 𝑚 ∑ 𝑖 = 𝑚 +1 𝑎 𝑖 𝑥 𝑖 + 𝑚 ∑ 𝑖 =3 𝑚 +1 (1 − 𝑎 𝑖 ) 𝑥 𝑖 with both maximums taken over all vectors 𝑎 ∈ {0 , 𝑚 . Since 𝑔 ( 𝑥 ) is just the sum max{ 𝑥 , 𝑥 𝑚 +1 } +max{ 𝑥 , 𝑥 𝑚 +2 } + ⋯ + max{ 𝑥 𝑚 , 𝑥 𝑚 } , and similarly for ℎ ( 𝑥 ) , the maximization problem 𝑓 can besolved using only 𝑚 = 𝑛 gates.It remains to show that 𝑓 indeed approximates the maximization problem on 𝐴 within factor 𝑟 = 2 .As we have shown in Section 6.2 (see Lemma 6.4), this happens precisely when the set 𝐵 lies below 𝐴 , and the set ⋅ 𝐴 lies below the convex hull Conv( 𝐵 ) of 𝐵 . It is clear that the first subset 𝐵 ′ of 𝐵 lies below 𝐴 . We have to show that this holds also for the second subset 𝐵 ′′ . For this, it is enough toshow that 𝐵 ′′ coincides with the set of all vectors (0 , 𝑎 , , 𝑎 ) for 𝑎 ∈ {0 , 𝑚 .It is known that a polynomial 𝑥 𝑘 permutes GF( 𝑞 ) if and only if 𝑞 − 1 and 𝑘 are relatively prime; see,for example, Lidl and Niederreiter [20, Theorem 7.8]. In our case, we have 𝑞 = 2 𝑚 and 𝑘 = 3 . Since 𝑚 is odd, we have 𝑚 = 2 𝑡 + 1 for some 𝑡 ∈ ℕ . Easy induction on 𝑡 shows that 𝑝 ( 𝑡 ) ∶= 2 𝑡 +1 + 1 is divisibleby : the basis 𝑡 = 0 is obvious, because 𝑝 (0) = 3 , and the induction step 𝑝 ( 𝑡 + 1) = 2 𝑡 +1)+1 + 1 =4(2 𝑡 +1 + 1) − 3 = 4 ⋅ 𝑝 ( 𝑡 ) − 3 follows from the induction hypothesis. So, 𝑞 − 1 = 𝑝 ( 𝑡 ) − 2 cannot bedivisible by , that is, 𝑞 − 1 and are relatively prime and, hence, the mapping 𝑎 ↦ 𝑎 is a bijection .This gives us a crucial fact: { (0 , 𝑎 , , 𝑎 ) ∶ 𝑎 ∈ {0 , 𝑚 } = { (0 , 𝑎, , 𝑎 ) ∶ 𝑎 ∈ {0 , 𝑚 } = 𝐵 ′′ . Hence, the entire set 𝐵 = 𝐵 ′ ∪ 𝐵 ′′ lies below 𝐴 , that is, every vector of 𝐵 is covered by at least onevector of 𝐴 . By Lemma 6.4, it remains to show that the set ⋅ 𝐴 lies below the convex hull Conv( 𝐵 ) .So, take an arbitrary vector 𝑢 = ⋅ ( 𝑎, 𝑎 , 𝑎, 𝑎 ) in ⋅ 𝐴 . This vector is a convex combination ⋅ 𝑣 + ⋅ 𝑤 of vectors 𝑣 = ( 𝑎, , 𝑎, and 𝑤 = (0 , 𝑎 , , 𝑎 ) of 𝐵 , as desired. (cid:3) A CKNOWLEDGMENTS
We thank Georg Schnitger and Igor Sergeev for inspiring discussions.R
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