Lower Bounds on Dynamic Programming for Maximum Weight Independent Set
LLower Bounds on Dynamic Programming forMaximum Weight Independent Set
Tuukka Korhonen ! ˇ Department of Computer Science, University of Helsinki, Finland
Abstract
We prove lower bounds on pure dynamic programming algorithms for maximum weight independentset (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and+ operations. For a graph G , an MWIS-circuit of G is a tropical circuit whose inputs correspondto vertices of G and which computes the weight of a maximum weight independent set of G forany assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d , then any MWIS-circuit of G has 2 Ω( w/d ) gates and that if G is planar then any MWIS-circuitof G has 2 Ω( w ) gates. An MWIS-formula is an MWIS-circuit where each gate has fan-out at mostone. We show that if G has treedepth t and maximum degree d , then any MWIS-formula of G has2 Ω( t/d ) gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and planar graphs, and treedepth characterizes optimal MWIS-formulasup to polynomials for all bounded degree graphs. Theory of computation → Graph algorithms analysis
Keywords and phrases
Maximum weight independent set, treewidth, tropical circuits, dynamicprogramming, treedepth, monotone circuit complexity
In this paper, we prove lower bounds for tropical circuits computing the weight of a maximumweight independent set (MWIS) of a graph. A tropical circuit is a circuit with Max and Plusoperations as gates. In particular, we consider MWIS-circuits of graphs. An MWIS-circuitof a graph G is a tropical circuit whose inputs correspond to the vertices of G and whichcomputes the weight of a maximum weight independent set of G for any assignment ofweights to the inputs. An MWIS-formula is an MWIS-circuit where each gate has fan-out atmost one.Our motivation for proving lower bounds for MWIS-circuits is that many algorithmictechniques for maximum weight independent set implicitly build an MWIS-circuit of the inputgraph, and therefore the running time of any algorithm resulting from such a technique isbounded from below by the minimum size of an MWIS-circuit of the input graph. Examplesof algorithmic techniques that build MWIS-circuits are dynamic programming over differentkinds of decompositions of graphs [3, 8, 14] and dynamic programming over potentialmaximal cliques [10, 15, 26]. Examples of algorithmic techniques that build MWIS-formulasare branching [18, 30] and maximal independent set enumeration [29]. Our results are unconditional lower bounds for sizes of MWIS-circuits and MWIS-formulasparameterized by graph parameters treewidth and treedepth, respectively. The lower boundsare exponential in treewidth and treedepth, and therefore well-known algorithms yieldmatching upper bounds for them [3, 16]. We emphasize that our lower bounds are notworst-case bounds, but instead hold for each individual graph, thus characterizing optimalMWIS-circuits and formulas for large classes of graphs. a r X i v : . [ c s . CC ] F e b Lower Bounds on Dynamic Programming for Maximum Weight Independent SetMWIS-circuits and Treewidth
We characterize optimal MWIS-circuits of bounded degree graphs. ▶ Theorem 1.
Any MWIS-circuit of a graph with treewidth w and maximum degree d has Ω( w/d ) gates. Theorem 1 is optimal up to a factor d in the sense that for each pair w, d we can constructa graph with treewidth Ω( w ) and maximum degree O ( d ) that admits an MWIS-formula with d w/d gates.A graph H is an induced minor of a graph G if it can be obtained from G by vertexdeletions and edge contractions. ▶ Theorem 2.
Let G be a graph that contains an induced minor with treewidth w andmaximum degree d . Any MWIS-circuit of G has Ω( w/ ( d d )) gates. Theorem 2 implies a 2 Ω( w ) lower bound for planar graphs because a planar graph withtreewidth w contains an Ω( w ) × Ω( w )-grid as an induced minor [33].The following corollary follows from Theorems 1 and 2, constant-factor treewidth approx-imation in poly ( n )2 O ( w ) time [32], and dynamic programming over a tree decomposition [3]. ▶ Corollary 3.
There is an algorithm which, given a bounded degree or planar graph G whosesmallest MWIS-circuit has τ gates, constructs an MWIS-circuit of G with poly ( τ ) gates in poly ( τ ) time. In particular, a property analogous to automatizability of proof systems [6] holds forMWIS-circuits on bounded degree graphs and planar graphs.
MWIS-formulas and Treedepth
We characterize optimal MWIS-formulas of bounded degree graphs. ▶ Theorem 4.
Any MWIS-formula of a graph with treedepth t and maximum degree d has Ω( t/d ) gates. Again, Theorem 4 is optimal up to a factor d by the same construction as Theorem 1.As formulas can be thought of as bounded space analogies of circuits, Theorem 4 givesfurther evidence (in addition to e.g. [9, 24, 31]) supporting that while treewidth is the rightparameter for CSP-like problems when equipped with unlimited space, treedepth is the rightparameter when dealing with bounded space.Obtaining a constant-factor single-exponential parameterized approximation algorithmfor treedepth is a well-known open problem [12], so while we know that the converse ofTheorem 4 existentially holds in bounded degree graphs, we currently do not know how toconstruct such MWIS-formulas without having the treedepth decomposition as an input. Our main circuit complexity tool is an adaptation of a circuit decomposition lemma usedin e.g. [20, 21, 34]. In particular, we show that this lemma can be adapted so that givenan MWIS-circuit with τ gates of a graph with treewidth w it extracts a family of τ vertexseparators each of size Ω( w ). Once this family has been extracted, the main challenge forproving Theorem 1 is to show that if this family of separators is too small, there exists anindependent set that intersects all of the separators. For this we use the lopsided Lovász uukka Korhonen 3 Local Lemma [13], though we note that more elementary arguments would suffice to provethe theorem with a worse dependency on d . To extend the result from bounded degree graphsto planar graphs we use a bounded degree induced minor with high treewidth to furthercontrol the structure of these separators.For MWIS-formulas parameterized by treedepth t we similarly extract a family of 2 τ vertex sets each of size Ω( t ) from a τ -gate MWIS-formula, showing that if an independentset intersects all of these vertex sets the formula cannot compute it. The same application ofthe Local Lemma is used to prove that such an independent set indeed exists in low degreegraphs if τ is too small. The argument for extracting the family from the formula is moread-hoc than the argument for circuits. The convention of modeling dynamic programming algorithms as tropical circuits originatesfrom the recent works of Jukna [22, 23], although some earlier results in monotone arithmeticcircuit complexity apply also to tropical circuits [21]. In general, tropical circuit lower boundsimply lower bounds for monotone arithmetic circuits, but not necessarily the other wayaround [22]. In addition to the works of Jukna, the other works explicitly giving lowerbounds for tropical circuits or formulas that we are aware of are [27, 28]. We are not awareof prior works on lower bounds for tropical circuits or formulas considering maximum weightindependent set or the parameters treewidth or treedepth.There are multiple worst-case hardness results related to different formulations of theindependent set polynomial. In [7] it was shown that the multivariate independent setpolynomial is VNP-complete. The univariate independent set polynomial is Ω( n/ log n ) worst-case complexity assuming Ω( w ) was shown for DNNF-compilation of monotone CNFswith primal treewidth w and bounded degree and arity, applying to all such CNFs [2]. Wenote that while this result superficially resembles our Theorem 1, lower bounds in the Booleansetting do not directly imply lower bounds for MWIS-circuits because the Boolean analogyof a polynomial computed by an MWIS-circuit is trivial W I ∈ IS ( G ) V v i ∈ I v i , where IS ( G )denotes the set of all independent sets, in particular including the empty set.Multiple worst-case lower bounds of form n Ω( w ) in limited models of computation forgraph homomorphism problems of a pattern graph with treewidth w to a graph with n vertices are known [4, 24, 25]. In particular, recently it was shown that the worst-casemonotone arithmetic circuit complexity of homomorphism polynomial is Θ( n w +1 ), and theworst-case arithmetic formula complexity is Θ( n t ), where t is the treedepth of the patterngraph [24]. In Section 2 we present preliminaries on graph theory, define MWIS-circuits and prove simplelemmas on them, and discuss the lopsided Lovász Local Lemma and prove a lemma using it.In Section 3 we prove the lower bounds for MWIS-circuits parameterized by treewidth, i.e.,Theorems 1 and 2. In Section 4 we prove the lower bound for MWIS-formulas parameterized
Lower Bounds on Dynamic Programming for Maximum Weight Independent Set by treedepth, i.e., Theorem 4. In Section 5 we give the construction that shows the optimalityof Theorems 1 and 4 up to a factor d . We conclude and discuss future work in Section 6. The vertex set of a graph G is denoted by V ( G ) and edge set by E ( G ). The set ofneighbors of a vertex v is denoted by N ( v ) and the neighborhood of a vertex set X by N ( X ) = S v ∈ X N ( v ) \ X . Closed neighborhoods are denoted by N [ v ] = N ( v ) ∪ { v } and N [ X ] = N ( X ) ∪ X . The subgraph G [ X ] induced by a vertex set X ⊆ V ( G ) has V ( G [ X ]) = X and E ( G [ X ]) = {{ u, v } ∈ E ( G ) | u ∈ X ∧ v ∈ X } . We also use G \ X = G [ V ( G ) \ X ] todenote induced subgraphs. An independent set is a vertex set I such that G [ I ] has no edges.A tree decomposition of a graph G is a tree T whose each vertex i ∈ V ( T ) correspondsto a bag B i ⊆ V ( G ), satisfying that V ( G ) = S B i , for each { u, v } ∈ E ( G ) there is a bag B i with { u, v } ⊆ B i , and for each v ∈ V ( G ) the subtree of T induced by bags containing v is connected.The width of a tree decomposition is max | B i | − tw ( G ) of a graph G isthe minimum width over its tree decompositions.A treedepth decomposition of a graph G is a rooted forest F with vertex set V ( F ) = V ( G ),satisfying for each { u, v } ∈ E ( G ) that u and v have an ancestor-descendant relation in F .The depth of F is the maximum number of vertices on a simple path from a root to a leaf in F .The treedepth td ( G ) of a graph G is the minimum depth over its treedepth decompositions.Note that tw ( G ) + 1 ≤ td ( G ). We start by giving a formal definition of tropical circuit. Our definition is non-standard inthat it does not allow any other input constants than 0, which we can w.l.o.g. assume inthe context of maximum weight independent set. For a comprehensive treatment of tropicalcircuits and their relations to monotone Boolean and monotone arithmetic circuits see [22]. ▶ Definition 5.
A tropical circuit over variables X is a directed acyclic graph with in-degreeof each vertex either 0 or 2. The vertices are called gates, the in-degree of a gate is calledfan-in, and the out-degree of a gate is called fan-out. Each gate with fan-in 0 is labeled witha variable x i ∈ X or the constant 0 and each gate with fan-in 2 is labeled with either max or+. One gate is designated as the output gate. A tropical formula is a tropical circuit whereeach gate has fan-out at most 1.With an assignment of real numbers to the variables X , a tropical circuit outputs anumber computed by the output gate by natural semantics, i.e., a gate labeled with a variable x i computes the value of x i , a gate labeled with 0 computes 0, a gate labeled with + computesthe sum of the values computed by its children, and a gate labeled with max computes themaximum of the values computed by its children. In particular, a tropical circuit computes atropical polynomial in the variables X over the tropical ( R ∪ {−∞} , max , +) semiring. In thetropical semiring max corresponds to addition and + corresponds to multiplication, with −∞ as the zero and 0 as the unit. We will refer to max as addition and to + as multiplication.We define an MWIS-polynomial with the following simple lemma. uukka Korhonen 5 ▶ Lemma 6.
Let G be a graph. A tropical circuit over variables V ( G ) computes the weightof a maximum weight independent set of G for any assignment of real weights to the inputsif and only if for the tropical polynomial f computed by the circuit it holds that each monomial of f is of form v · . . . · v l , where { v , . . . , v l } is an independent set of G and for each independent set { v , . . . , v l } of G there is a monomial v · . . . · v l in f , includingthe empty independent set corresponding to the empty product . Proof.
For the if-direction, (1) guarantees that the value computed by the circuit is at mostthe weight of a maximum weight independent set and (2) guarantees that the value is atleast the weight of a maximum weight independent set.For the only if-direction, if some monomial would not be multilinear, i.e., include a factor v for some vertex v , the output would be incorrect when assigning weight 1 to v and 0 toother vertices. If some monomial would be of form v · . . . · v l , where { v , . . . , v l } is not anindependent set the output would be incorrect when assigning weight 1 to those v i and 0 toothers. Finally, if the output polynomial would not include v · . . . · v l as a monomial forsome independent set { v , . . . , v l } then the circuit would be incorrect when assigning weight1 to vertices of this independent set and − ◀ An MWIS-polynomial of a graph G is a polynomial f satisfying (1) and (2) in Lemma 6.An MWIS-circuit of G is a tropical circuit that computes an MWIS-polynomial of G . AnMWIS-formula of G is an MWIS-circuit of G that is a tropical formula.We note that requiring the circuit to work for all real weights is not a strong assumption:Any MWIS-circuit that works for weights { , } can be turned into an MWIS-circuit thatworks for weights R ∪ {−∞} by replacing each input variable v i by max( v i , ▶ Definition 7.
A partial MWIS-polynomial is a polynomial f satisfying (1) in Lemma 6. Apartial MWIS-circuit is a tropical circuit computing a partial MWIS-polynomial.Note that by monotonicity of (max , +) computations we can assume that each gate ofan MWIS-circuit computes a partial MWIS-polynomial and therefore each subcircuit is apartial MWIS-circuit. ▶ Definition 8.
Let f be a partial MWIS-polynomial. We denote by Sup( f ) the support of f , that is, the variables that occur in the monomials of f .We also use Sup( g ) for a gate g to denote the support of the polynomial computed bythe gate. Note that each monomial of f corresponds to an independent set of G [Sup( f )].The following property is the basis for proving lower bounds for MWIS-circuits. ▶ Lemma 9.
Let f = g · h be a partial MWIS-polynomial of a graph G . The sets N [ Sup ( g )] and Sup ( h ) are disjoint. Proof.
If there was a vertex v ∈ Sup( g ) ∩ Sup( h ) then f would contain a monomial with afactor v . If there was a vertex v ∈ Sup( g ) and u ∈ Sup( h ) with { u, v } ∈ E ( G ), then therewould be a monomial in f containing a factor u · v . ◀ We will say that a partial MWIS-polynomial f or a circuit computing f computes anindependent set I if f contains the monomial Q v i ∈ I v i . In particular, an MWIS-polynomialcomputes every independent set. Lower Bounds on Dynamic Programming for Maximum Weight Independent Set
The lopsided Lovász Local Lemma [13] (see [1] for the general version) is a method forshowing that there is a non-zero probability that none of the events in a collection of eventshold. In particular, we use it to show that independent sets satisfying certain requirementsexist. ▶ Definition 10.
Let E , . . . , E n be events in a probability space. A graph Γ is a negativedependency graph of the events if its vertices are V (Γ) = {E , . . . , E n } and for all events E i and subsets J ⊆ V (Γ) \ N ( E i ) it holds that Pr[ S j ∈ J E j | E i ] ≥ Pr[ S j ∈ J E j ].In words, the negative dependency graph should capture all negative correlations betweenthe events. ▶ Proposition 11 ([1]) . Let E , . . . , E n be a collection of events with a negative dependencygraph Γ . If there exists real numbers x , . . . , x n with ≤ x i ≤ such that for each i it holdsthat Pr[ E i ] ≤ x i Q E j ∈ N ( E i ) (1 − x j ) , then Pr[ T ni =1 E i ] > . We prove a lemma which captures our use of the Local Lemma in Theorems 1 and 4. Wespell out the constants to emphasize that they are not particularly high, although notingthat a more careful proof could improve them a bit. ▶ Lemma 12.
Let G be a graph with maximum degree d and F a family of vertex subsets of G , each member of F containing at least k vertices. If |F| ≤ e k/ (6 d ) , then there exists anindependent set of G that intersects all sets in F . Proof.
We assume d ≥ d ≤
1. We use the Local Lemmato construct such an independent set. We let each vertex be in the independent set withprobability p = 1 / (2 d ). Our bad events are E e for each each edge e indicating that bothendpoints of e are selected in the independent set, and E A for each A ∈ F indicating thatthe independent set does not intersect A . The negative dependency graph is a bipartitegraph connecting E e to E A if at least one of the endpoints of e is in A . In particular, notethat the edge events E e have non-negative correlation with each other and the vertex setevents E A also have non-negative correlation with each other. For all edge events E e wechoose x e = 1 / (3 d + 1) and for all vertex set events E A we choose x A = 1 / (5 |F| + 1). Nowit suffices to verify thatPr[ E e ] = p ≤ x e (1 − x A ) |F| (1)andPr[ E A ] = (1 − p ) | A | ≤ x A (1 − x e ) | A | d (2)hold whenever 6 |F| ≤ e | A | / d .For (1), a lower bound for the right hand side is e − / / (3 d + 1), which can be verified tobe greater than p = 1 / (4 d ) when d ≥
2. For (2), an upper bound for the left hand side is e −| A | / (2 d ) , and a lower bound for the right hand side is x A e −| A | d/ (3 d ) , implying that (2) holdsif e −| A | / (2 d ) e | A | / (3 d ) ≤ x A . This simplifies to e −| A | / (6 d ) ≤ x A ⇔ e | A | / (6 d ) ≥ |F| + 1. ◀ uukka Korhonen 7 In this section we prove lower bounds for MWIS-circuits parameterized by treewidth, i.e.,Theorems 1 and 2.We use a witness of high treewidth due to Robertson-Seymour approximation algorithm [32].A separation of a graph G is a pair of vertex sets ( A, B ) such that A ∪ B = V ( G ) and novertex of A \ B is adjacent to a vertex of B \ A . The order of a separation ( A, B ) is | A ∩ B | . Aseparation ( A, B ) is a balanced separation of a vertex set X ⊆ V ( G ) if | ( A \ B ) ∩ X | ≤ | X | / | ( B \ A ) ∩ X | ≤ | X | / ▶ Lemma 13 ([32]) . If a graph G has treewidth at least k , then there is a vertex set X ⊆ V ( G ) such that any balanced separation of X in G has order at least k . The next lemma is our main tool to connect circuit complexity with treewidth. Thislemma is an adaptation of a classical circuit decomposition lemma (e.g. Theorem 1 in [20],Lemma 3 in [34]). In our applications the vertex set X will be the set given by Lemma 13. ▶ Lemma 14.
Let G be a graph and X ⊆ V ( G ) with | X | ≥ . If there is an MWIS-circuitof G with τ gates, then we can write an MWIS-polynomial of G as g · h + . . . + g τ · h τ ,where for all i it holds that | Sup ( g i ) ∩ X | ≤ | X | / and | Sup ( h i ) ∩ X | ≤ | X | / . Proof.
Let f + e be an MWIS-polynomial of G , where f can be computed by a tropicalcircuit with τ gates. We will show that there is an MWIS-polynomial f ′ + g · h + e of G ,where f ′ can be computed by a tropical circuit with τ − | Sup( g ) ∩ X | ≤ | X | / | Sup( h ) ∩ X | ≤ | X | /
3. The lemma follows from this by induction.If | Sup( f ) ∩ X | ≤ | X | / f down starting from the output gate, always choosing the one of the two child gates whosesupport has larger intersection with X , until we reach a gate v computing a polynomial f v with | X | / ≤ | Sup( f v ) ∩ X | ≤ | X | /
3. Let f v = −∞ be the polynomial computed by thecircuit when the value of the gate v is set to −∞ . Now we can write an MWIS-polynomial of G as f v = −∞ + f v · g + e , for example by letting g be an MWIS-polynomial of G \ N [Sup( f v )].Now, we observe that f v = −∞ can be computed by a circuit with τ − f v and g cannot intersect, and therefore | Sup( g ) ∩ X | ≤ | X | / ◀ Now we complete the proof of Theorem 1 by putting Lemmas 12, 13, and 14 together. ▶ Lemma 15.
Let G be a graph with maximum degree d and treewidth at least k . AnyMWIS-circuit of G has at least e k/ (6 d ) / gates. Proof.
Suppose there is an MWIS-circuit of G with τ gates. By Lemma 13 there is a vertexset X ⊆ V ( G ) that does not admit a balanced separation of order less than k . By Lemma 14we can write an MWIS-polynomial of G as g · h + . . . + g τ · h τ , where for all i it holds that | Sup( g i ) ∩ X | ≤ | X | / | Sup( h i ) ∩ X | ≤ | X | /
3. Now, by Lemma 9 each multiplication g i · h i defines a balanced separation ( V ( G ) \ Sup( h i ) , V ( G ) \ Sup( g i )) of X . The order of sucha separation is | V ( G ) \ Sup( g i · h i ) | , and therefore | V ( G ) \ Sup( g i · h i ) | ≥ k . Note that g i · h i does not compute an independent set I if I intersects V ( G ) \ Sup( g i · h i ). Therefore, by letting F be the collection of vertex sets { V ( G ) \ Sup( g · h ) , . . . , V ( G ) \ Sup( g τ · h τ ) } , Lemma 12shows that if 6 τ ≤ e k/ (6 d ) we can construct an independent set that is not computed by anyof the multiplications, contradicting the assumption that we have an MWIS-circuit. ◀ Lower Bounds on Dynamic Programming for Maximum Weight Independent Set
Note that the sizes of minimum MWIS-circuits are clearly monotone with respect to inducedsubgraphs, so we can without loss of generality replace “induced minor” by “contraction” inthe statement of Theorem 2.A contraction model of a graph H in a graph G is a function f : V ( H ) → V ( G ) , where2 V ( G ) denotes the power set of V ( G ), satisfying that { f ( v ) | v ∈ V ( H ) } is a partition of V ( G ), for each v ∈ V ( H ) the induced subgraph G [ f ( v )] is connected, and { u, v } ∈ E ( H ) if and only if N ( f ( u )) intersects f ( v ).A graph G contains a graph H as a contraction if and only if there is a contraction modelof H in G . For v ∈ V ( H ) we call the induced subgraphs G [ f ( v )] clusters .First, we ensure that the maximum degree of each cluster is bounded. ▶ Lemma 16.
Let G be a graph that contains a graph H with maximum degree d as acontraction. There is an induced subgraph G ′ of G that has a contraction model f of H sothat the maximum degree of each cluster G ′ [ f ( v )] is at most d . Proof.
Consider a contraction model f of H in G and a cluster G [ f ( v )] for some v ∈ V ( H ).Because the degree of H is at most d , we can assign the cluster a set of at most d terminalvertices whose connectivity should be preserved in order to satisfy that f is a contractionmodel of H in G . Now, we can remove from the cluster any vertices as long as the terminalsstay connected. In particular, if there is a vertex u with degree > d in G [ f ( v )], then wecan consider the shortest paths from u to the terminals, and remove the neighbors of u in G [ f ( v )] that do not participate in the shortest paths, thus making the degree of u in G [ f ( v )]at most d . ◀ We also need the following lemma. ▶ Lemma 17.
Let I be an independent set selected uniformly at random from the set ofall independent sets of a graph G with maximum degree d . For all v ∈ V ( G ) it holds that Pr[ v ∈ I ] ≥ / d +1 . Proof.
For any set J ⊆ N ( v ) it holds that Pr[ I ∩ N ( v ) = J ] ≤ Pr[ I ∩ N ( v ) = ∅ ] becausewe can map any independent set I with I ∩ N ( v ) = J into an independent set I \ N ( v ).Therefore Pr[ I ∩ N ( v ) = ∅ ] ≥ / d , so by observing that Pr[ v ∈ I | I ∩ N ( v ) = ∅ ] ≥ / v ∈ I ] ≥ / d +1 . ◀ Next we finish the proof with similar arguments as in the proof of Theorem 1, but with adifferent kind of construction of the independent set with the Local Lemma. In this case theconstants involved appear to be impractical. ▶ Lemma 18.
Let G be a graph that contains a graph H with maximum degree d andtreewidth k as a contraction. Any MWIS-circuit of G has Ω( k/ ( d d )) gates. Proof.
Let f be the contraction model of H in G . First, by Lemma 16 we can assume thatthe maximum degree of each cluster G [ f ( v )] is at most d . Now, by Lemma 13 we let X ′ be avertex set of H that has no balanced separation of order less than k . Then we let X be avertex set of G created by mapping each v ∈ X ′ to an element of f ( v ). For each balancedseparation ( A, B ) of X in G , the set A ∩ B must intersect at least k different clusters, becauseotherwise we could map it into a balanced-separation of X ′ of order < k in H . Therefore, byassuming that G has an MWIS-circuit with τ gates and applying Lemma 14 with the set X uukka Korhonen 9 we write an MWIS-polynomial of G as g · h + . . . + g τ · h τ , observing that for each i the set S i = V ( G ) \ Sup( g i · h i ) intersects at least k different clusters. Now it remains to show thatif τ is too small we can construct an independent set of G that intersects S i for all i .By removing vertices from each S i we can assume that S i contains exactly one vertexfrom each cluster that it intersects. We use the Local Lemma to construct the independentset. First we select each cluster independently with probability p = 1 / (4 d d ), and then foreach selected cluster G [ f ( v )] we select an independent set uniformly at randomly from the setof all independent sets of G [ f ( v )]. By Lemma 17 each vertex will appear in the independentset with probability at least p/ d +1 . Vertices in different clusters appear in it independentlyof each other.Now our bad events are E { u,v } for all { u, v } ∈ E ( H ) indicating that both clusters G [ f ( u )]and G [ f ( v )] have been selected and E i for each S i indicating that the set S i does not intersectthe independent set. Our negative dependency graph has edges connecting each E { u,v } toeach E i such that S i intersects f ( u ) or f ( v ). It also has all edges between all events E i because E i and E j can be negatively correlated if S i and S j intersect a common cluster.For edges { u, v } ∈ E ( H ) we let x { u,v } = 1 / (15 d d + 1) and for sets S i we choose x i = 1 / (20 τ + 1). Now it suffices to verify thatPr[ E { u,v } ] = p ≤ x { u,v } (1 − x i ) τ (3)andPr[ E i ] ≤ (1 − p/ d +1 ) | S i | ≤ x i (1 − x i ) τ (1 − x { u,v } ) | S i | d (4)hold whenever 30 τ ≤ e | S i | / (120 d d ) . We also assume that d ≥ d ≤ H is at most 2.For (3), a lower bound for the right hand side is e − / / (15 d d + 1), which is greaterthan p = 1 / (16 d d ) when d ≥
2. For (4), a lower bound for the right hand sideis x i e − / e −| S i | d/ (15 d d ) and an upper bound for the left hand side is e −| S i | / (8 d d ) , soit holds whenever e −| S i | / (8 d d ) ≤ x i e − / e −| S i | d/ (15 d d ) holds, which we can simplify to e | S i | (1 / (15 d d ) − / (8 d d )) ≤ x i e − / , and finally to e − | S i | / (120 d d ) ≤ / (20 τ + 1) e − / , whichholds whenever 30 τ ≤ e | S i | / (120 d d ) . ◀ For treedepth we are not aware of linear high-treedepth witnesses similar to Lemma 13.However, it turns out that we can use very basic properties of treedepth decompositions toestablish the connection to formula complexity.Recall that we denote the treedepth of a graph G with td ( G ). The following propertiesfollow from the definition of treedepth. ▶ Proposition 19.
Let G be a graph with treedepth td ( G ) . It holds that td ( G \ { v } ) ≥ td ( G ) − for any v ∈ V ( G ) and td ( G ) is the maximum of td ( G [ C ]) over the connected components C of G . For our proof we need to introduce two definitions on MWIS-formulas. We start bydefining typical independent sets of a partial MWIS-formulas. ▶ Definition 20.
Let F be a partial MWIS-formula of a graph G . An independent set I of G isa typical independent set of F if for each multiplication gate g with td ( G [Sup( g )]) ≥ td ( G ) / I intersects a connected component C of G [Sup( g )] with td ( G [ C ]) = td ( G [Sup( g )]). Note that by the property 2 of Proposition 19 such component indeed exists.We also define the separator
Sep( g ) of a gate g . ▶ Definition 21.
The separator of the output gate o is Sep( o ) = V ( G ) \ Sup( o ). The separatorof a gate g whose parent p is a multiplication gate is Sep( g ) = Sep( p ). The separator of agate g whose parent p is a sum gate is Sep( g ) = Sep( p ) ∪ Sup( p ) \ Sup( g ).With the definitions of typical independent sets and separators of gates, we can state thefollowing lemma which will be applied with Lemma 12 to prove our lower bound. ▶ Lemma 22.
Let G be a graph with td ( G ) ≥ and F a partial MWIS-formula of G . If I isa typical independent set of F and intersects Sep ( g ) for each gate g with | Sep ( g ) | ≥ td ( G ) / ,then F does not compute I . Proof.
Let F be such a formula and I such an independent set. We say that a gate g of F isredundant if F computes I if and only if F without g computes I . First, note that all gates g such that I intersects Sep( g ) are redundant because by the definition of separator thereis an ancestor gate g ′ of g with a sum gate parent p such that none of the monomials M contributed from g ′ to p have M = Q v i ∈ I ∩ Sup( p ) v i , implying that g ′ is redundant and thusall of its descendants are redundant.Now, we prove by induction starting from the leaves that every gate g of F for which | Sep( g ) | + td ( G [Sup( g )]) ≥ td ( G ) holds is redundant. First, for all such gates g with td ( G [sup( g )]) ≤ td ( G ) /
2, including all leaves, we have that | Sep( g ) | ≥ td ( G ) /
2, making g redundant by our definition of I . For a sum gate g and its child c we have by property 1of Proposition 19 that td ( G [Sup( c )]) ≥ td ( G [Sup( g )]) − | Sup( g ) \ Sup( c ) | , rearranging to td ( G [Sup( c )] ≥ td ( G [Sup( g )]) − | Sep( c ) | + | Sep( g ) | , and finally to td ( G [Sup( c )]) + | Sep( c ) | ≥ td ( G [Sup( g )]) + | Sep( g ) | . This implies that if | Sep( g ) | + td ( G [Sup( g )]) ≥ td ( G ) then bothchildren of g are redundant, making g redundant. For a multiplication gate g with td ( G [Sup( g )]) ≥ td ( G ) / c of g with td ( G [Sup( c )]) + | Sep( c ) | = td ( G [Sup( g )]) + | Sep( g ) | such that g is redundant if c is redundant. Therefore the induction works, and because | Sep( o ) | + td ( G [Sup( o )]) ≥ td ( G )holds for the output gate o , the output gate is redundant and therefore the formula does notcompute I . ◀ Now the only thing left to complete the proof of Theorem 4 is to show that if a formulahas less than 2 Ω( td ( G ) /d ) gates then we can construct an independent set that is typical forthe formula and intersects Sep( g ) whenever | Sep( g ) | ≥ td ( G ) /
2. For an independent setto be typical it suffices that it intersects Sup( g ) for all gates g with | Sup( g ) | ≥ td ( G ) / F consisting of Sep( g ) for all | Sep( g ) | ≥ td ( G ) / g ) for all | Sup( g ) | ≥ td ( G ) /
2. This yields a lower bound of e td ( G ) / (12 d ) /
12 for thenumber of gates.
We show that for each pair w, d we can construct a graph with treewidth Ω( w ) and maximumdegree O ( d ) that admits an MWIS-formula with d w/d gates.If d > w then a d -clique does the job. Otherwise, we take a bounded degree expanderwith w/d vertices, having treewidth Ω( w/d ), constructible by e.g. [17]. We replace eachvertex of the expander with a d -clique (which will be referred to as cluster) such that eachvertex of a cluster is connected to each vertex of the clusters of the adjacent vertices. Wedenote the constructed graph with G w,d uukka Korhonen 11 ▶ Proposition 23.
The graph G w,d has treewidth Ω( w ) , maximum degree O ( d ) , and admitsan MWIS-formula with d w/d gates. Proof.
The maximum degree is clearly ( d + 1) times the maximum degree of the originalbounded degree expander. The treewidth is Ω( w ) because if a balanced separator containsone vertex from a cluster it must contain all vertices of the cluster.Note that by a simple recursion any n -vertex graph admits an MWIS-formula with atmost 2 n gates, so the original expander admits an MWIS-formula with 2 w/d gates. We canconstruct an MWIS-formula of G w,d by taking the MWIS-formula of the original expanderand replacing each leaf corresponding to a vertex v with d -gate construction computing themaximum over the vertices of the cluster of v . ◀ We investigated the tropical circuit complexity of maximum weight independent set. Ourinitial motivation for this was the fact that lower bounds for tropical circuits imply lowerbounds for many actual algorithmic techniques for maximum weight independent set thatare widely used in both theory and practice. We showed that in bounded degree graphsoptimal MWIS-circuits are characterized by treewidth and optimal MWIS-formulas arecharacterized by treedepth. We generalized the result for MWIS-circuits to apply beyondbounded degree graphs, to a graph class that includes all planar graphs. The constantshidden by the Ω-notation in our results are somewhat practical, even though we did notspecifically optimize them. For example, Theorem 1 shows that any MWIS-circuit of the5000 × gates.We identify some technical barriers for extending the results. First, we note that Lemma 12is not effective in graphs with maximum degree higher than k : If | N ( v ) | ≥ k , we can add N ( v ) to F to force the independent set to avoid v , essentially forcing us to work with G \ { v } .Indeed an example of a graph with high treewidth and no small MWIS-circuits for whichLemma 12 is unsuitable is a clique with each edge subdivided. In some cases, including thesubdivided clique and planar graphs, this barrier can be circumvented with Theorem 2 byusing bounded degree induced minors with high treewidth. We also note that our proofs donot exploit the fact that the separators given by Lemma 14 are balanced beyond just thesize bound. By ad-hoc analysis this balance can be used to show the lower bound for thesubdivided clique.An interesting general direction for future work could be to prove Corollary 3 for aslarge graph classes as possible. As the natural next step from our results we see boundeddegeneracy graphs, as for them an exponential lower bound parameterized by treewidthwould seem natural because each induced subgraph contains an exponential number ofindependent sets. We hope that this line of work will lead to new insights on the structureof independent sets that could even be useful for positive results on algorithms for maximumweight independent set. References Noga Alon and Joel H. Spencer.
The Probabilistic Method, Third Edition . Wiley-Interscienceseries in discrete mathematics and optimization. Wiley, 2008. Antoine Amarilli, Florent Capelli, Mikaël Monet, and Pierre Senellart. Connecting knowledgecompilation classes and width parameters.
Theory Comput. Syst. , 64(5):861–914, 2020. doi:10.1007/s00224-019-09930-2 . Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problemsrestricted to partial k-trees.
Discrete Applied Mathematics , 23(1):11–24, 1989. doi:10.1016/0166-218X(89)90031-0 . Per Austrin, Petteri Kaski, and Kaie Kubjas. Tensor network complexity of multilinear maps.In , volume 124 of
LIPIcs , pages 7:1–7:21. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2019. doi:10.4230/LIPIcs.ITCS.2019.7 . Markus Bläser and Christian Hoffmann. On the complexity of the interlace polynomial. In
STACS 2008, 25th Annual Symposium on Theoretical Aspects of Computer Science, Bordeaux,France, February 21-23, 2008, Proceedings , volume 1 of
LIPIcs , pages 97–108. Schloss Dagstuhl- Leibniz-Zentrum fuer Informatik, Germany, 2008. doi:10.4230/LIPIcs.STACS.2008.1337 . Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. On interpolation and automatization forfrege systems.
SIAM J. Comput. , 29(6):1939–1967, 2000. doi:10.1137/S0097539798353230 . Irénée Briquel and Pascal Koiran. A dichotomy theorem for polynomial evaluation. In
Math-ematical Foundations of Computer Science 2009, 34th International Symposium, MFCS 2009,Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009. Proceedings , volume 5734 of
LectureNotes in Computer Science , pages 187–198. Springer, 2009. doi:10.1007/978-3-642-03816-7\_17 . Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Boolean-width of graphs.
Theor.Comput. Sci. , 412(39):5187–5204, 2011. doi:10.1016/j.tcs.2011.05.022 . Li-Hsuan Chen, Felix Reidl, Peter Rossmanith, and Fernando Sánchez Villaamil. Width, depth,and space: Tradeoffs between branching and dynamic programming.
Algorithms , 11(7):98,2018. doi:10.3390/a11070098 . Maria Chudnovsky, Marcin Pilipczuk, Michal Pilipczuk, and Stéphan Thomassé. On themaximum weight independent set problem in graphs without induced cycles of length at leastfive.
SIAM J. Discret. Math. , 34(2):1472–1483, 2020. doi:10.1137/19M1249473 . Vasek Chvátal. Determining the stability number of a graph.
SIAM J. Comput. , 6(4):643–662,1977. doi:10.1137/0206046 . Wojciech Czerwinski, Wojciech Nadara, and Marcin Pilipczuk. Improved bounds for theexcluded-minor approximation of treedepth. In , volume 144of
LIPIcs , pages 34:1–34:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. doi:10.4230/LIPIcs.ESA.2019.34 . Paul Erdös and Joel Spencer. Lopsided Lovász local lemma and latin transversals.
Discret.Appl. Math. , 30(2-3):151–154, 1991. doi:10.1016/0166-218X(91)90040-4 . Fedor V. Fomin and Petr A. Golovach. Subexponential parameterized algorithms and kerneliz-ation on almost chordal graphs. In
ESA 2020 , volume 173 of
LIPIcs , pages 49:1–49:17. SchlossDagstuhl - Leibniz-Zentrum für Informatik, 2020. doi:10.4230/LIPIcs.ESA.2020.49 . Fedor V. Fomin, Ioan Todinca, and Yngve Villanger. Large induced subgraphs via triangulationsand CMSO.
SIAM Journal on Computing , 44(1):54–87, 2015. doi:10.1137/140964801 . Eugene C. Freuder and Michael J. Quinn. Taking advantage of stable sets of variables inconstraint satisfaction problems. In
Proceedings of the 9th International Joint Conferenceon Artificial Intelligence. Los Angeles, CA, USA, August 1985 , pages 1076–1078. MorganKaufmann, 1985. URL: http://ijcai.org/Proceedings/85-2/Papers/082.pdf . Ofer Gabber and Zvi Galil. Explicit constructions of linear-sized superconcentrators.
J.Comput. Syst. Sci. , 22(3):407–420, 1981. doi:10.1016/0022-0000(81)90040-4 . Peter Gartland and Daniel Lokshtanov. Independent set on P k -free graphs in quasi-polynomialtime. In FOCS 2020 , pages 613–624. IEEE, 2020. doi:10.1109/FOCS46700.2020.00063 . Christian Hoffmann. Exponential time complexity of weighted counting of independent sets. In
Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai,India, December 13-15, 2010. Proceedings , volume 6478 of
Lecture Notes in Computer Science ,pages 180–191. Springer, 2010. doi:10.1007/978-3-642-17493-3\_18 . uukka Korhonen 13 Laurent Hyafil. On the parallel evaluation of multivariate polynomials.
SIAM J. Comput. ,8(2):120–123, 1979. doi:10.1137/0208010 . Mark Jerrum and Marc Snir. Some exact complexity results for straight-line computationsover semirings.
J. ACM , 29(3):874–897, 1982. doi:10.1145/322326.322341 . Stasys Jukna. Lower bounds for tropical circuits and dynamic programs.
Theory Comput.Syst. , 57(1):160–194, 2015. doi:10.1007/s00224-014-9574-4 . Stasys Jukna. Tropical complexity, sidon sets, and dynamic programming.
SIAM J. Discret.Math. , 30(4):2064–2085, 2016. doi:10.1137/16M1064738 . Balagopal Komarath, Anurag Pandey, and C. S. Rahul. Graph homomorphism polynomials:Algorithms and complexity.
CoRR , abs/2011.04778, 2020. URL: https://arxiv.org/abs/2011.04778 , arXiv:2011.04778 . Yuan Li, Alexander A. Razborov, and Benjamin Rossman. On the AC complexity of subgraphisomorphism. SIAM J. Comput. , 46(3):936–971, 2017. doi:10.1137/14099721X . Daniel Lokshantov, Martin Vatshelle, and Yngve Villanger. Independent set in P SODA 2014 , pages 570–581. SIAM, 2014. doi:10.1137/1.9781611973402.43 . Meena Mahajan, Prajakta Nimbhorkar, and Anuj Tawari. Computing the maximum using (min,+) formulas. In , volume 83 of
LIPIcs , pages74:1–74:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. doi:10.4230/LIPIcs.MFCS.2017.74 . Meena Mahajan, Prajakta Nimbhorkar, and Anuj Tawari. Shortest path length with bounded-alternation (min, +) formulas.
International Journal of Advances in Engineering Sciences andApplied Mathematics , 11(1):68–74, 2019. J. W. Moon and L. Moser. On cliques in graphs.
Israel Journal of Mathematics , 3(1):23–28,1965. Patric R. J. Östergård. A fast algorithm for the maximum clique problem.
Discret. Appl.Math. , 120(1-3):197–207, 2002. doi:10.1016/S0166-218X(01)00290-6 . Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structuraldecompositions of graphs.
ACM Trans. Comput. Theory , 9(4):18:1–18:36, 2018. doi:10.1145/3154856 . Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width.
J.Algorithms , 7(3):309–322, 1986. doi:10.1016/0196-6774(86)90023-4 . Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a planar graph.
J.Comb. Theory, Ser. B , 62(2):323–348, 1994. doi:10.1006/jctb.1994.1073 . Leslie G. Valiant. Negation can be exponentially powerful.
Theor. Comput. Sci. , 12:303–314,1980. doi:10.1016/0304-3975(80)90060-2doi:10.1016/0304-3975(80)90060-2