On the Power and Limitations of Branch and Cut
Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson
aa r X i v : . [ c s . CC ] F e b On the Power and Limitations of Branch and Cut
Noah Fleming Mika G¨o¨os † Russell Impagliazzo
UniversityofToronto EPFL UniversityofCalifornia,&SimonsInstitute SanDiego
Toniann Pitassi Robert Robere † Li-Yang Tan
UniversityofToronto&IAS McGillUniversity StanfordUniversity
Avi Wigderson
IAS
February 10, 2021
Abstract
The Stabbing Planes proof system [8] was introduced to model the reasoning carried out inpractical mixed integer programming solvers. As a proof system, it is powerful enough to sim-ulate Cutting Planes and to refute the Tseitin formulas — certain unsatisfiable systems of linearequations mod2 — which are canonical hard examples for many algebraic proof systems. Ina recent (and surprising) result, Dadush and Tiwari [21] showed that these short refutations ofthe Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planesproofs, refuting a long-standing conjecture. This translation raises several interesting ques-tions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes.This would allow for the substantial analysis done on the Cutting Planes system to be lifted topractical mixed integer programming solvers. Second, whether the quasi-polynomial depth ofthese proofs is inherent to Cutting Planes.In this paper we make progress towards answering both of these questions. First, we showthat any
Stabbing Planes proof with bounded coefficients ( SP ∗ ) can be translated into CuttingPlanes. As a consequence of the known lower bounds for Cutting Planes, this establishes thefirst exponential lower bounds on SP ∗ . Using this translation, we extend the result of Dadushand Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system oflinear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari,our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this isinherent. As a step towards this conjecture, we develop a new geometric technique for provinglower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lowerbounds on the depth of Semantic
Cutting Planes proofs of the Tseitin formulas. † Work done while at Institute for Advanced Study.
Introduction
An effective method for analyzing classes of algorithms is to formalize the techniques used by the classinto a formal proof system , and then analyze the formal proof system instead. By doing this, theoristsare able to hide many of the practical details of implementing these algorithms, while preserving theclass of methods that the algorithms can feasibly employ. Indeed, this approach has been applied tostudy many different families of algorithms, such as•
Conflict-driven clause-learning algorithms for SAT [37, 45, 53], which can be formalized using resolution proofs [23].• Optimization algorithms using semidefinite programming [30,47], which can often be formalizedusing
Sums-of-Squares proofs [6, 34].• The classic cutting planes algorithms for integer programming [17, 31], which are formalized by cutting planes proofs [17, 20].In the present work, we continue the study of formal proof systems corresponding to modern integerprogramming algorithms. Recall that in the integer programming problem, we are given a polytope P ⊆ R n and a vector c ∈ R n , and our goal is to find a point x ∈ P ∩ Z n maximizing c · x . Theclassic approach to solving this problem — pioneered by Gomory [31] — is to add * cutting planes to P . A cutting plane for P is any inequality of the form ax ≤ ⌊ b ⌋ , where a is an integral vector, b isrational, and every point of P is satisfied by ax ≤ b . By the integrality of a , it follows that cuttingplanes preserve the integral points of P , while potentially removing non-integral points from P . Thecutting planes algorithms then proceed by heuristically choosing “good” cutting planes to add to P totry and locate the integral hull of P as quickly as possible.As mentioned above, these algorithms can be naturally formalized into a proof system — the Cut-ting Planes proof system , denoted CP — as follows [20]. Initially, we are given a polytope P , presentedas a list of integer-linear inequalities { a i x ≤ b i } . From these inequalities we can then deduce new in-equalities using two deduction rules:• Linear Combination.
From inequalities ax ≤ b, cx ≤ d , deduce any non-negative linear combi-nation of these two inequalities with integer coefficients.• Division Rule.
From an inequality ax ≤ b , if d ∈ Z divides all entries of a then deduce ( a/d ) x ≤⌊ b/d ⌋ .A Cutting Planes refutation of P is a proof of the trivially false inequality ≤ from the inequalitiesin P ; clearly, such a refutation is possible only if P does not contain any integral points. While CuttingPlanes has grown to be an influential proof system in propositional proof complexity, the originalcutting planes algorithms suffered from numerical instabilities, as well as difficulties in finding goodheurisitics for the next cutting planes to add [31].The modern algorithms in integer programming improve on the classical cutting planes methodby combining them with a second technique, known as branch-and-bound , resulting in a family ofoptimization algorithms broadly referred to as branch-and-cut algorithms . These algorithms search forinteger solutions in a polytope P by recursively repeating the following two procedures: First, P issplit into smaller polytopes P , . . . , P k such that P ∩ Z n ⊆ S i ∈ [ k ] P i (i.e. branching ). Next, cutting * Throughout, we will say that a cutting plane, or an inequality is added to a polytope P to mean that it is added to the setof inequalities defining P . cutting ). In practice,branching is usually performed by selecting a variable x i and branching on all possible values of x i ; thatis, recursing on P ∩ { x i = t } for each feasible integer value t . More complicated branching schemeshave also been considered, such as branching on the hamming weight of subsets of variables [27],branching using basis-reduction techniques [1, 2, 41], and more general linear inequalities [38, 43, 46].However, while these branch-and-cut algorithms are much more efficient in practice than the clas-sical cutting planes methods, they are no longer naturally modelled by Cutting Planes proofs. So, inorder to model these solvers as proof systems, Beame et al. [8] introduced the Stabbing Planes proofsystem. Given a polytope P containing no integral points, a Stabbing Planes refutation of P proceedsas follows. We begin by choosing an integral vector a , an integer b , and replacing P with the twopolytopes P ∩ { ax ≤ b − } and P ∩ { ax ≥ b } . Then, we recurse on these two polytopes, continuinguntil all descendant polytopes are empty (that is, they do not even contain any real solutions). The ma-jority of branching schemes used in practical branch-and-cut algorithms (including all of the concreteschemes mentioned above) are examples of this general branching rule.It is now an interesting question how the two proof systems — Cutting Planes and Stabbing Planes— are related. By contrasting the two systems we see at least three major differences:• Top-down vs. Bottom-up.
Stabbing Planes is a top-down proof system, formed by performingqueries on the polytope and recursing; while Cutting Planes is a bottom-up proof system, formedby deducing new inequalities from old ones.•
Polytopes vs. Halfspaces.
Individual “lines” in a Stabbing Planes proof are polytopes , whileindividual “lines” in a Cutting Planes proof are halfspaces .• Tree-like vs. DAG-like.
The graphs underlying Stabbing Planes proofs are trees, while the graphsunderlying Cutting Planes proofs are general DAGs: intuitively, this means that Cutting Planesproofs can “re-use” their intermediate steps, while Stabbing Planes proofs cannot.When taken together, these facts suggest that Stabbing Planes and Cutting Planes could be incompara-ble in power, as polytopes are more expressive than halfspaces, while DAG-like proofs offer the powerof line-reuse. Going against this natural intuition, Beame et al. proved that Stabbing Planes can actuallyefficiently simulate Cutting Planes [8]. Furthermore, they proved that Stabbing Planes is equivalent tothe proof system tree-like R ( CP ) , denoted TreeR ( CP ) , which was introduced by Kraj´ıˇcek [40], andwhose relationship to Cutting Planes was previously unknown.This leaves the converse problem — of whether Stabbing Planes can also be simulated by CuttingPlanes — as an intriguing open question. Beame et al. conjectured that such a simulation was impossi-ble, and furthermore that the Tseitin formulas provided a separation between these systems [8]. For anygraph G and any { , } -labelling ℓ of the vertices of G , the Tseitin formula of ( G, ℓ ) is the followingsystem of F -linear equations: for each edge e we introduce a variable x e , and for each vertex v wehave an equation M u : uv ∈ E x uv = ℓ ( v ) asserting that the sum of the edge variables incident with v must agree with its label ℓ ( v ) (note sucha system is unsatisfiable as long as P v ℓ ( v ) is odd). On the one hand, Beame et al. proved that thereare quasi-polynomial size Stabbing Planes refutations of the Tseitin formulas [8]. On the other hand,Tseitin formulas had long been conjectured to be exponentially hard for Cutting Planes [20], as theyform one of the canonical families of hard examples for algebraic and semi-algebraic proof systems,including Nullstellensatz [33], Polynomial Calculus [16], and Sum-of-Squares [34, 51].2n a recent breakthrough, the long-standing conjecture that Tseitin was exponentially hard for Cut-ting Planes was refuted by Dadush and Tiwari [21], who gave quasi-polynomial size
Cutting Planesrefutations of Tseitin instances. Moreover, to prove their result, Dadush and Tiwari showed how to translate the quasipolynomial-size Stabbing Planes refutations of Tseitin into Cutting Planes refuta-tions. This translation result is interesting for several reasons. First, it brings up the possibility thatCutting Planes can actually simulate Stabbing Planes. If possible, such a simulation would allow thesignificant analysis done on the Cutting Planes system to be lifted directly to branch-and-cut solvers. Inparticular, this would mean that the known exponential-size lower bounds for Cutting Planes refutationswould immediately imply the first exponential lower bounds for these algorithms for arbitrary branch-ing heuristics. Second, the translation converts shallow
Stabbing Planes proofs into very deep
CuttingPlanes proofs: the Stabbing Planes refutation of Tseitin has depth O (log n ) and quasi-polynomialsize, while the Cutting Planes refutation has quasipolynomial size and depth. This is quite unusualsince simulations between proof systems typically preserve the structure of the proofs, and thus bringsup the possibility that the Tseitin formulas yield a supercritical size/depth tradeoff – formulas withshort proofs, requiring superlinear depth. For contrast: another simulation from the literature whichemphatically does not preserve the structure of proofs is the simulation of bounded-size resolution by bounded-width resolution by Ben-Sasson and Wigderson [10]. In this setting, it is known that thissimulation is tight [13], and even that there exist formulas refutable in resolution width w requiringmaximal size n Ω( w ) [5]. Furthermore, under the additional assumption that the proofs are tree-like ,Razborov [50] proved a supercritical trade-off between width and size. A New Characterization of Cutting Planes
Our first main result gives a characterization of Cutting Planes proofs as a natural subsystem of Stab-bling Planes that we call facelike
Stabbing Planes. A Stabbing Planes query is facelike if one of the sets P ∩ { ax ≤ b − } or P ∩ { ax ≥ b } is either empty or is a face of the polytope P , and a Stabbing Planesproof is said to be facelike if it only uses facelike queries. Our main result is the following theorem. Theorem 1.1.
The proof systems CP and Facelike SP are polynomially equivalent. Using this equivalence we prove the following surprising simulation, stating that Stabbing Planesproofs with relatively small coefficients (quasi-polynomially bounded in magnitude) can be quasi-polynomially simulated by Cutting Planes.
Theorem 1.2.
Let F be any unsatisfiable CNF formula on n variables, and suppose that there is a SP refutation of F in size s and maximum coefficient size c . Then there is a CP refutation of F in size s ( cn ) log s . As a second application of Theorem 1.1, we generalize Dadush and Tiwari’s result to show thatCutting Planes can refute any unsatisfiable system of linear equations over a finite field. This followsby showing that, like Tseitin, we can refute such systems of linear equations in quasipolynomial-sizeFacelike SP . Theorem 1.3.
Let F be the CNF encoding of an unsatisfiable system of m linear equations over a finitefield. There is a CP refutation of F of size | F | O (log m ) . This should be contrasted with the work of Filmus, Hrubeˇs, and Lauria [26], which gives severalunsatisfiable systems of linear equations over R that require exponential size refutations in CuttingPlanes. 3 ower Bounds An important open problem is to prove superpolynomial size lower bounds for Stabbing Planes proofs.We make significant progress toward this goal by proving the first superpolynomial lower bounds onthe size of low-weight Stabbing Planes proofs. Let SP ∗ denote the family of Stabbing Planes proofs inwhich each coefficient has at most quasipolynomial ( n log O (1) n ) magnitude. Theorem 1.4.
There exists a family of unsatisfiable CNF formulas { F n } such that any SP ∗ refutationof F requires size at least n ε for constant ε > . Our proof follows straightforwardly from Theorem 1.2 together with known Cutting Planes lowerbounds. We view this as a step toward proving SP lower bounds (with no restrictions on the weight).Indeed, lower bounds for CP ∗ (low-weight Cutting Planes) [14] were first established, and led to (un-restricted) CP lower bounds [48].Our second lower bound is a new linear depth lower bound for semantic Cutting Planes proofs. (Ina semantic Cutting Planes proof the deduction rules for CP are replaced by a simple and much stronger semantic deduction rule .) Theorem 1.5.
For all sufficiently large n there is a graph G on n vertices and a labelling ℓ such thatthe Tseitin formula for ( G, ℓ ) requires Ω( n ) depth to refute in Semantic Cutting Planes. We note that lower bounds for Semantic Cutting Planes were already established via communica-tion complexity arguments. However, since Tseitin formulas have short communication protocols, ourdepth bound for semantic Cutting Planes proofs of Tseitin is new.Our main motivation behind this depth bound is as a step towards proving a supercritical tradeoff.A supercritical tradeoff for CP , roughly speaking, states that efficient CP proofs must sometimes bevery deep — that is, beyond the trivial depth upper bound of O ( n ) [11, 50]. Establishing supercriticaltradeoffs is a major challenge, both because hard examples witnessing such a tradeoff are rare, andbecause current methods seem to fail beyond the critical regime. In fact, the only supercritical tradeoffbetween size and depth established to date is for bounded-width, tree-like Resolution [50].As we mentioned above, Dadush and Tiwari’s quasipolynomial-size CP refutations of Tseitin arequasipolynomially deep, and similarly our simulation of Facelike Stabbing Planes by Cutting Planes inTheorem 1.1 is similarly far from depth-preserving. We therefore conjecture that the Tseitin formulasexhibit a supercritical tradeoff for CP . Our proof of Theorem 1.5 is a novel geometric argument whichgeneralizes the top-down “protection lemma” approach [15] that crucially relied on the exact deductionrules of CP. As our argument has no inherent barrier for going behind the critical regime, we hope thatit is a step towards proving a supercritical tradeoff, which we leave as an open problem. Conjecture 1.6.
There exists a family of unsatisfiable formulas { F n } such that F n has quasipolynomial-size CP proofs, but any quasipolynomial-size proof requires superlinear depth. Lower Bounds on SP and TreeR ( CP ) . Several lower bounds on subsystems of SP and TreeR ( CP ) have already been established. Kraj´ıˇcek [40] proved exponential lower bounds on the size of R ( CP ) proofs in which both the width of the clauses and the magnitude of the coefficients of each line in theproof are bounded. Concretely, let these bounds be w and c respectively. The lower bound that he ob-tains is n Ω(1) /c w log n . Kojevnikov [39] removed the dependence on the coefficient size for TreeR ( CP ) P Semantic
CPCP = Facelike
SPSP ∗ CP ∗ Figure 1: Known relationships between proof systems considered in this paper. A solid black (red)arrow from proof system P to P indicates that P can polynomially (quasi-polynomially) simulate P . A dashed arrow indicates that this simulation cannot be done.proofs, obtaining a bound of exp(Ω( p n/w log n )) . Beame et al. [8] provide a size-preserving simula-tion of Stabbing Planes by TreeR ( CP ) which translates a depth d Stabbing Planes proof into a width d TreeR ( CP ) proof, and therefore this implies lower bounds on the size of SP proofs of depth o ( n/ log n ) .Beame et al. [8] exhibit a function for which there are no SP refutations of depth o ( n/ log n ) via areduction to the communication complexity of the CNF search problem. Supercritical Tradeoffs.
Besides the work of Razborov [50], a number of supercritical tradeoffshave been observed in proof complexity, primarily for proof space. Beame et al. [7] and Beck et al. [9]exhibited formulas which admit polynomial size refutations in Resolution and the Polynomial Calculusrespectively, and such that any refutation of sub-linear space necessitates a superpolynomial blow-upin size. Recently, Berkholz and Nordstr¨om [11] gave a supercritical trade-off between width and spacefor Resolution.
Depth in Cutting Planes and Stabbing Planes.
It is widely known (and easy to prove) that anyunsatisfiable family of CNF formulas can be refuted by exponential size and linear depth CuttingPlanes. It is also known that neither Cutting Planes nor Stabbing Planes can be balanced , in the sensethat a depth- d proof can always be transformed into a size O ( d ) proof [8, 15]. This differentiates bothof these proof systems from more powerful proof systems like Frege, for which it is well-known howto balance arbitrary proofs [19]. Furthermore, even though both the Tseitin principles and systemsof linear equations in finite fields can be proved in both quasipolynomial-size and O (log n ) depth inFacelike SP , the simulation of Facelike SP by CP cannot preserve both size and depth, as the Tseitinprinciples are known to require depth Θ( n ) to refute in CP [15].We first recall the known depth lower bound techniques for both Cutting Planes and StabbingPlanes proofs. In both proof systems, arguably the primary method for proving depth lower bounds isby reducing to real communication complexity [8, 36] ; however, communication complexity is alwaystrivially upper bounded by n , and it is far from clear how to use the assumption on the size of the proofto boost this to superlinear.A second method has been developed for proving lower bounds (applicable to Cutting Planes butnot to Stabbing Planes) using so-called protection lemmas [15], which seems much more amenable toapplying a small-size assumption on the proof. We also remark that for many formulas (such as theTseitin formulas!) it is known how to achieve Ω( n ) -depth lower bounds in Cutting Planes via protectionlemmas, while proving even ω (log n ) lower bounds via communication complexity is impossible, dueto a known folklore upper bound. 5 Preliminaries
We first recall the definitions of some key proof systems.
Resolution.
Fix an unsatisfiable CNF formula F over variables x , . . . , x n . A Resolution refutation P of F is a sequence of clauses { C i } i ∈ [ s ] ending in the empty clause C s = ∅ such that each C i is in F or is derived from earlier clauses C j , C k with j, k < i using one of the following rules:• Resolution. C i = ( C j \ { ℓ k } ) ∪ ( C k \ (cid:8) ℓ k (cid:9) ) where ℓ k ∈ C j , ℓ k ∈ C k is a literal.• Weakening. C i ⊇ C j .The size of the resolution proof is s , the number of clauses. It is useful to visualize the refutation P asa directed acyclic graph; with this in mind the depth of the proof (denoted depth Res ( P ) ) is the lengthof the longest path in the proof DAG. The resolution depth depth Res ( F ) of F is the minimal depth ofany resolution refutation of F . Cutting Planes and Semantic Cutting Planes. A Cutting Planes proof ( CP ) of an inequality cx ≥ d from a system of linear inequalities P is given by a sequence of inequalities a x ≥ b , a x ≥ b , . . . , a s x ≥ b s such that a s = c , b s = d , and each inequality a i x ≥ b i is either in P or is deduced from earlierinequalities in the sequence by applying one of the two rules Linear Combination or Division Rule described at the beginning of Section 1. We will usually be interested in the case that the list ofinequalities P defines a polytope.An alternative characterization of Cutting Planes uses Chv´atal-Gomory cuts (or just
CG cuts ) [17,20]. Let P be a polytope. A hyperplane ax = b is supporting for P if b = max { ax : x ∈ P } , andif ax = b is a supporting hyperplane then the set P ∩ { x ∈ R n : ax = b } is called a face of P . Aninequality ax ≤ b is valid for P if every point of P satisfies the inequality and ax = b is a supportinghyperplane of P . Definition 2.1.
Let P ⊆ R n be a polytope, and let ax ≥ b be any valid inequality for P such that allcoefficients of a are relatively prime integers. The halfspace { x ∈ R n : ax ≥ ⌈ b ⌉} is called a CG cut for P . (We will sometimes abuse notation and refer to the inequality ax ≥ ⌈ b ⌉ also as a CG cut.)If ax ≥ ⌈ b ⌉ is a CG cut for the polytope P , then we can derive ax ≥ ⌈ b ⌉ from P in O ( n ) steps ofCutting Planes by Farkas Lemma (note that the inequality ax ≥ b is valid for P by definition, so wecan deduce ax ≥ b as a linear combination of the inequalities of P and then apply the division rule). If P is a polytope and H is a CG cut, then we will write P ⊢ P ∩ H , and say that P ∩ H is derived from P . Given a CNF formula F , we can translate F into a system of linear inequalities in the followingnatural way. First, for each variable x i in F add the inequality ≤ x i ≤ . If C = W i ∈ P x i ∨ W i ∈ N ¬ x i is a clause in F , then we add the inequality X i ∈ P x i + X i ∈ N (1 − x i ) ≥ .
6t is straightforward to see that the resulting system of inequalities will have no integral solutions if andonly if the original formula F is unsatisfiable. With this translation we consider Cutting Planes refuta-tions (defined in the introduction) of F to be refutations of the translation of F to linear inequalities.The semantic Cutting Planes proof system (denoted sCP or Semantic CP ) is a strengthening ofCutting Planes proofs to allow any deduction that is sound over integral points [14]. Like CuttingPlanes, an sCP proof is given by a sequence of halfspaces { a i x ≥ c i } i ∈ [ s ] , but now we can use thefollowing very powerful semantic deduction rule :• Semantic Deduction.
From a j x ≥ c j and a k x ≥ c k deduce a i x ≥ c i if every { , } solution of a i x ≥ c i is also an integral solution of both a j x ≥ c j and a k x ≥ c k .Filmus et al. [26] showed that sCP is extremely strong: there are instances for which any refutationin CP requires exponential size, and yet these instances admit polynomial-size refutations in semantic sCP .The size of a Cutting Planes proof is the number of lines (it is known that for unsatisfiable CNFformulas that this measure is polynomially related to the length of the bit-encoding of the proof [20]).As with Resolution, it is natural to arrange Cutting Planes proofs into a proof DAG. With this inmind we analogously define depth CP ( F ) and depth sCP ( F ) to be the smallest depth of any (semantic)Cutting Planes proof of F .It is known that any system of linear inequalities in the unit cube has CP depth at most O ( n log n ) ,and moreover there are examples requiring CP -depth more than n [25]. However for unsatisfiable CNFformulas, the CP -depth is at most n [12]. Stabbing Planes.
Let F be an unsatisfiable system of linear inequalities. A Stabbing Planes ( SP )refutation of F is a directed binary tree, T , where each edge is labelled with a linear integral inequalitysatisfying the following consistency conditions :• Internal Nodes.
For any internal node u of T , if the right outgoing edge of u is labelled with ax ≥ b , then the left outgoing edge is labelled with its integer negation ax ≤ b − .• Leaves.
Each leaf node v of T is labelled with a non-negative linear combination of inequalitiesin F with inequalities along the path leading to v that yields ≥ .For an internal node u of T , the pair of inequalities ( ax ≤ b − , ax ≥ b ) is called the query corre-sponding to the node. Every node of T has a polytope P associated with it, where P is the polytopedefined by the intersection of the inequalities in F together with the inequalities labelling the path fromthe root to this node. We will say that the polytope P corresponds to this node. The slab correspondingto the query is { x ∗ ∈ R n | b − < ax ∗ < b } , which is the set of points ruled out by this query. The width of the slab is the minimum distance between ax ≤ b − and ax ≥ b , which is / k a k . The size of a refutation is the bit-length needed to encode a description of the entire proof tree, which, forCNF formulas as well as sufficiently bounded systems of inequalities, is polynomially equivalent to thenumber of queries in the refutation [21]. As well, the depth of the refutation is the depth of the binarytree. The proof system SP ∗ is the subsystem of Stabbing Planes obtained by restricting all coefficientsof the proofs to have magnitude at most quasipolynomial ( n log O (1) n ) in the number of input variables.The Stabbing Planes proof system was introduced by Beame et al. [8] as a generalization of CuttingPlanes that more closely modelled query algorithms and branch-and-bound solvers. Beame et al. provedthat SP is equivalent to the proof system TreeR ( CP ) introduced by Kraj´ıˇcek [40] which can be thoughtof as a generalization of Resolution where the literals are replaced with integer-linear inequalities.7 Translating Stabbing Planes into Cutting Planes CP with Subsystems of SP In this section we prove Theorem 1.1, restated below, which characterizes Cutting Planes as a non-trivial subsystem of Stabbing Planes.
Theorem 1.1.
The proof systems CP and Facelike SP are polynomially equivalent. We begin by formally defining Facelike SP . Definition 3.1.
A Stabbing Planes query ( ax ≤ b − , ax ≥ b ) at a node P is facelike if one of the sets P ∩ { x ∈ R n : ax ≤ b − } , P ∩ { x ∈ R n : ax ≥ b } is empty or a face of P (see Figure 2b). An SP refutation is facelike if every query in the refutation is facelike.Enroute to proving Theorem 1.1, it will be convenient to introduce the following further restrictionof Facelike Stabbing Planes. Definition 3.2.
A Stabbing Planes query ( ax ≤ b − , ax ≥ b ) at a node corresponding to a polytope P is pathlike if at least one of P ∩ { x ∈ R n : ax ≤ b − } and P ∩ { x ∈ R n : ax ≥ b } is empty (seeFigure 2a). A Pathlike SP refutation is one in which every query is pathlike.The name “pathlike” stems from the fact that the underlying graph of a pathlike Stabbing Planesproof is a path, since at most one child of every node has any children (see Figure 2). In fact, we havealready seen (nontrivial) pathlike SP queries under another name: Chv´atal-Gomory cuts. Lemma 3.3.
Let P be a polytope and let ( ax ≤ b − , ax ≥ b ) be a pathlike Stabbing Planes queryfor P . Assume w.l.o.g. that P ∩ { x ∈ R n : ax ≤ b − } = ∅ and that P ∩ { x ∈ R n : ax ≥ b } ( P .Then ax ≥ b is a CG cut for P .Proof. Since ax ≥ b is falsified by some point in P , it follows that there exists some < ε < suchthat ax ≥ b − ε is valid for P — note that ε < since otherwise ax ≤ b − would not have emptyintersection with P . This immediately implies that ax ≥ b is a CG cut for P .With this observation we can easily prove that Pathlike SP is equivalent to CP . Throughout theremainder of the section, for readability, we will use the abbreviation P ∩ { ax ≥ b } for P ∩ { x ∈ R n : ax ≥ b } , for any polytope P and linear inequality ax ≥ b . Lemma 3.4.
Pathlike SP is polynomially equivalent to CP .Proof. First, let a x ≥ b , a x ≥ b , . . . , a s x ≥ b s be a CP refutation of an unsatisfiable systemof linear inequalities Ax ≥ b . Consider the sequence of polytopes P = { Ax ≥ b } and P i = P i − ∩ { a i x ≥ b i } . By inspecting the rules of CP , it can observed that P i ∩ { a i x ≤ b i − } = ∅ andthus P i +1 can be deduced using one pathlike SP query from P i for all ≤ i ≤ s .Conversely, let P be any polytope and let ( ax ≤ b − , ax ≥ b ) be any pathlike SP query to P (so, suppose w.l.o.g. that the halfspace defined by ax ≤ b − has empty intersection with P ). ByLemma 3.3, ax ≥ b is a CG cut for P , and so can be deduced in Cutting Planes from the inequalitiesdefining P in length O ( n ) (cf. Section 2). Applying this to each query in the Pathlike SP proof yieldsthe theorem. 8 x ≥ bax ≤ b − ∅ ∅ ax ≤ b − ax ≥ bP (a) A Pathlike query. The polytope P ∩ { x ∈ R n : ax ≤ b − } = ∅ , and ax ≥ b is a CG cut for P . ax ≥ bax ≤ b − ax ≤ b − ax ≥ bPax = b − (b) A Facelike query. The polytope P ∩ { x ∈ R n : ax ≤ b − } = P ∩ { x ∈ R n : ax = b } is a face of P . Figure 2: Pathlike and Facelike SP queries on a polytope P . On the left are the proofs and on the rightare the corresponding effects on the polytope.Next, we show how to simulate Facelike SP proofs by Pathlike SP proofs of comparable size. Theproof of Lemma 3.6 is inspired by Dadush and Tiwari [21], and will use the following lemma due toSchrijver [52] (although, we use the form appearing in [20]). Recall that we write P ⊢ P ′ for polytopes P, P ′ to mean that P ′ can be obtained from P by adding a single CG cut to P . Lemma 3.5 (Lemma 2 in [20]) . Let P be a polytope defined by a system of integer linear inequalitiesand let F be a face of P . If F ⊢ F ′ then there is a polytope P ′ such that P ⊢ P ′ and P ′ ∩ F ⊆ F ′ . Lemma 3.6.
Facelike SP is polynomially equivalent to Pathlike SP .Proof. That Facelike SP simulates Pathlike SP follows by the fact that any Pathlike SP query is avalid query in Facelike SP . For the other direction, consider an SP refutation π of size t . We describe arecursive algorithm for generating a Pathlike SP proof from π . The next claim will enable our recursivecase. Claim.
Let P be a polytope and suppose ax ≥ b is valid for P . Assume that P ∩ { ax = b } has aPathlike SP refutation using s queries. Then P ∩ { ax ≥ b + 1 } can be derived from P in Pathlike SP using s + 1 queries. Proof of Claim.
Since ax ≥ b is valid for P it follows that F = P ∩ { ax = b } is a face of P by definition. Consider the Pathlike SP refutation F , F , . . . F s = ∅ , where the i th polytope F i for i < s is obtained from F i − by applying a pathlike SP query and proceeding to the non-empty child.Without loss of generality we may assume that F i ( F i − for all i , and so applying Lemma 3.39e have that F i − ⊢ F i for all i . Thus, by applying Lemma 3.5 repeatedly, we get a sequence ofpolytopes P = P ⊢ P ⊢ · · · ⊢ P s such that P i ∩ F = P i ∩ { ax = b } ⊆ F i . This means that P s ∩ { ax = b } ⊆ F s = ∅ , and so ( ax ≤ b, ax ≥ b + 1) is Pathlike SP query for P s . This meansthat P s ⊢ P s ∩ { ax ≥ b + 1 } ⊆ P ∩ { ax ≥ b + 1 } . Since any CG cut can be implemented as aPathlike SP query the claim follows by applying the s CG cuts as pathlike queries, followed by thequery ( ax ≤ b, ax ≥ b + 1) .We generate a Pathlike SP refutation by the following recursive algorithm, which performs an in-order traversal of π . At each step of the recursion (corresponding to a node in π ) we maintain thecurrent polytope P we are visiting and a Pathlike SP proof Π — initially, P is the initial polytope and Π = ∅ . We maintain the invariant that when we finish the recursive step at node P , the Pathlike SP refutation Π is a refutation of P . The algorithm is described next:1. Let ( ax ≤ b − , ax ≥ b ) be the current query and suppose that ax ≥ b − is valid for P .2. Recursively refute P ∩ { ax ≤ b − } = P ∩ { ax = b − } , obtaining a Pathlike SP refutation Π with t queries.3. Apply the above Claim to deduce P ∩ { ax ≥ b } from P in t + 1 queries.4. Refute P ∩ { ax ≥ b } by using the SP refutation for the right child.Correctness follows immediately from the Claim, and also since the size of the resulting proof is thesame as the size of the SP refutation.Theorem 1.1 then follows by combining Lemma 3.4 with Lemma 3.6. SP ∗ by CP In this section we prove Theorem 1.2, restated below for convenience.
Theorem 1.2.
Let F be any unsatisfiable CNF formula on n variables, and suppose that there is a SP refutation of F in size s and maximum coefficient size c . Then there is a CP refutation of F in size s ( cn ) log s . To prove this theorem, we will show that any low coefficient SP proof can be converted into aFacelike SP proof with only a quasi-polynomial loss. If P is a polytope let d ( P ) denote the diameter of P , which is the maximum Euclidean distance between any two points in P . Theorem 1.2 followsimmediately from the following theorem. Theorem 3.7.
Let P be a polytope and suppose there is an SP refutation of P with size s and maximumcoefficient size c . Then there is a Facelike SP refutation of P in size s ( c · d ( P ) √ n ) log s . Proof.
The theorem is by induction on s . Clearly, if s = 1 then the tree is a single leaf and the theoremis vacuously true.We proceed to the induction step. Let P be the initial polytope and π be the SP proof. Considerthe first query ( ax ≤ b, ax ≥ b + 1) made by the proof, and let π L be the SP proof rooted at the leftchild (corresponding to ax ≤ b ) and let π R be the SP proof rooted at the right child. Let P L denote the10olytope at the left child and P R denote the polytope at the right child. By induction, let π ′ L and π ′ R bethe Facelike SP refutations for P L and P R guaranteed by the statement of the theorem.Suppose w.l.o.g. that | π L | ≤ | π | / . Let b be the largest integer such that ax ≥ b is satisfied forany point in P . The plan is to replace the first query ( ax ≤ b, ax ≥ b + 1) with a sequence of queries q , q , . . . , q t − such that• For each i , q i = ( ax ≤ b + i, ax ≥ b + i + 1) .• The query q is the root of the tree and q i is attached to the right child of q i − for i ≥ .• q t − = ( ax ≤ b, ax ≥ b + 1) .After doing this replacement, instead of having two child polytopes P L , P R below the top query, wehave t + 1 polytopes P , P , . . . , P t +1 where P i = P ∩ { ax = b + i } and P t +1 = P R . To finish theconstruction, for each i ≤ t use the proof π ′ L to refute P i and the proof π ′ R to refute P t +1 .We need to prove three statements: this new proof is a valid refutation of P , the new proof isfacelike, and that the size bound is satisfied.First, it is easy to see that this is a valid proof, since for each i ≤ t the polytope P i ⊆ P L and P t +1 ⊆ P R — thus, the refutations π ′ L and π ′ R can be used to refute the respective polytopes.Second, to see that the proof is facelike, first observe that all the queries in the subtrees π ′ L , π ′ R arefacelike queries by the inductive hypothesis. So, we only need to verify that the new queries at the topof the proof are facelike queries, which can easily be shown by a quick induction. First, observe thatthe query q is a facelike query, since b was chosen so that ax ≥ b is valid for the polytope P . Byinduction, the query q i = ( ax ≤ b + i, ax ≥ b + i + 1) is a facelike query since the polytope P i associated with that query is P ∩ { ax ≥ b + i } by definition. Thus ax ≥ b + i is valid for the polytopeat the query.Finally, we need to prove the size upper bound. Let s be the size of the original proof, s L be thesize of π L and s R be the size of π R . Observe that the size of the new proof is given by the recurrencerelation f ( s ) = t · f ( s L ) + f ( s R ) . where f (1) = 1 . Since the queries q , q , . . . , q t − cover the polytope P L with slabs of width / k a k ,it follows that t ≤ d ( P L ) k a k ≤ d ( P ) √ n k a k ∞ = d ( P ) c √ n where we have used that the maximum coefficient size in the proof is c . Thus, by induction, the previousinequality, and the assumption that s L ≤ s/ , we can conclude that the size of the proof is f ( s ) ≤ s L ( c · d ( P ) √ n )( c · d ( P L ) √ n ) log s L + s R ( c · d ( P R ) √ n +) log s R ≤ s L ( c · d ( P ) √ n )( c · d ( P ) √ n ) log( s/ + s R ( c · d ( P ) √ n ) log s ≤ s L ( c · d ( P ) √ n ) log s + s R ( c · d ( P ) √ n ) log s = s ( c · d ( P ) √ n ) log s . Theorem 1.2 follows immediately, since for any CNF formula F the encoding of F as a system oflinear inequalities is contained in the n -dimensional cube [0 , n , which has diameter √ n . We may alsoimmediately conclude Theorem 1.4 by applying the known lower bounds on the size of Cutting Planesproofs [28, 29, 35, 48].As a consequence of Theorem 1.2 and the non-automatability of Cutting Planes [32], we can con-clude that SP ∗ proofs cannot be found efficiently assuming P = NP . Indeed, non-automatability of SP ∗ follows by observing that the argument [32] does not require large coefficients.11 Refutations of Linear Equations over a Finite Field
In this section we prove Theorem 1.3. To do so, we will extend the approach used by Beame et al. [8]to prove quasi-polynomial upper bounds on the Tseitin formulas to work on any unsatisfiable set oflinear equations over any finite field.If ax = b is a linear equation we say the width of the equation is the number of non-zero variablesoccurring in it. Any width- d linear equation over characteristic q can be represented by a CNF formulawith q d − width- d clauses — one ruling out each falsifying assignment. For a width- d system of m linear equations F over a finite field, we will denote by | F | := mq d − the size of the CNF formulaencoding F . Theorem 4.1.
Let F = { f = b , . . . , f m = b m } be a width- d , unsatisfiable set of linear equationsover characteristic q . There is an SP refutation of (the CNF encoding of) F in size ( mqd ) O (log m ) q d = | F | O (log m ) . First we sketch the idea over characteristic . In this case the SP proof corresponds to a branchdecomposition procedure which is commonly used to solve SAT (see e.g. [3, 22, 24, 42]). View thesystem F as a hypergraph over n vertices (corresponding to the variables) and with a d -edge for eachequation. Partition the set of hyperedges into two sets E = E ∪ E of roughly the same size, andconsider the cut of vertices that belong to both an edge in E and in E . Using the SP rule we branchon all possible values of the sum of the cut variables in order to isolate E and E . Once we know thissum, we are guaranteed that either E is unsatisfiable or E is unsatisfiable depending on the parityof the of the sum of the cut variables. This allows us to recursively continue on the side of the cut( E or E ) that is unsatisfiable. Since there are n Boolean variables, each cut corresponds to at most n + 1 possibilities for the sum, and if we maintain that the partition of the hyper edges defining the cutis balanced, then we will recurse at most O (log m ) times. This gives rise to a tree decomposition offanout O ( n ) and height O (log n ) .Over characteristic q the proof will proceed in much the same way. Instead of a subgraph, at eachstep we will maintain a subset of the equations I ⊆ [ m ] such that { f i = b i } i ∈ I must contain a constraintthat is violated by the SP queries made so far. We partition I into two sets I and I of roughly equalsize and query the values a and b of P i ∈ I f i and P i ∈ I f i . Because F is unsatisfiable, at least one of a − P i ∈ I b i or b − P i ∈ I b i , meaning that that it is unsatisfiable, and we recurse on it.In the following, we will let z stand for a vector of F q -valued variables z i . When we discuss anyform f := az where a ∈ F mq and z is a vector of n variables z i , we will implicitly associate it with thelinear form P i ∈ [ n ] a i ( P j ∈ [log q ] x i,j ) where x i,j are the log q many Boolean variables encoding z i inthe CNF encoding of F . Proof of Theorem 4.1.
Let F = { f = b , . . . , f m = b m } be a system of unsatisfiable linear equationsover F q , where each f i = a i z for a i ∈ F nq , and b i ∈ F q . Because F is unsatisfiable, there exists a F q linear combination of the equations in F witnessing this; formally, there exists α ∈ F nq such that P i ∈ [ m ] α i f i ≡ q , but P i ∈ [ m ] α i b i q .Stabbing Planes will implement the following binary search procedure for a violated equation;we describe the procedure first, and then describe how to implement it in Stabbing Planes. In eachround we maintain a subset I ⊆ [ m ] and an integer k I representing the value of P i ∈ I α i f i . Over thealgorithm, we maintain the invariant that k I − P i ∈ I b i q , which implies that there must be acontradiction to F inside of the constraints { f i = b i } i ∈ I .12nitially, I = [ m ] and we obtain k I by querying the value of the sum P i ∈ [ m ] α i f i . If k I q then this contradicts the fact that P i ∈ I α i f i ≡ q ; thus, the invariant holds. Next,perform the following algorithm.1. Choose a balanced partition I = I ∪ I (so that || I | − | I || ≤ ).2. Query the value of P i ∈ I α i f i and P i ∈ I α i f i ; denote these values by a and b respectively.3. If a − P i ∈ I α i b i q then recurse on I with k I := a . Otherwise, if b − P i ∈ I α i b i q then recurse on I with k I := b .4. Otherwise (if a − P i ∈ I α i b i ≡ b − P i ∈ I α i b i ≡ q ), then this contradicts the invariant: k I − X i ∈ I b i = X i ∈ I α i ( f i − b i )= X i ∈ I α i ( f i − b i ) + X i ∈ I α i ( f i − b i )= ( a − X i ∈ I α i b i ) + ( b − X i ∈ I α i b i ) ≡ q. This recursion stops when | I | = 1 , at which point we have an immediate contradiction between k I andthe single equation indexed by I .It remains to implement this algorithm in SP . First, we need to show how to perform the queries instep 2. Querying the value of any sum P i ∈ I α i f i can be done in a binary tree with at most q md leaves,one corresponding to every possible query outcome. Internally, this tree queries all possible integervalues for this sum (e.g. ( P i ∈ I α i f i ≤ , P i ∈ I α i f i ≥ , ( P i ∈ I α i f i ≤ , P i ∈ I α i f i ≥ , . . . ). Forthe leaf where we have deduced P i ∈ [ m ] α i f i ≤ we use the fact that each variable is non-negative todeduce that P i ∈ [ m ] α i f i ≥ as well. Note that q md is an upper bound on this sum because there are m equations, each containing at most d variables, each taking value at most ( q − † . Thus, step 2 canbe completed in ( q md ) queries.Finally, we show how to derive refutations in the following cases: (i) when we deduced that P i ∈ [ m ] α i f i q at the beginning, (ii) in step 4, (iii) when | I | = 1 .(i) Suppose that we received the value a q from querying P i ∈ [ m ] α i f i . Note that everyvariable in P i ∈ [ m ] α i f i is a multiple of q . Query (cid:16) X i ∈ [ m ] α i f i /q ≤ ⌈ a/q ⌉ − , X i ∈ [ m ] α i f i /q ≥ ⌈ a/q ⌉ (cid:17) . At the leaf that deduces P i ∈ [ m ] α i f i /q ≤ ⌈ a/q ⌉ − , we can derive ≥ as a non-negativelinear combination of this inequality together with P i ∈ [ m ] α i f i ≥ a . Similarly, at the other leaf P i ∈ [ m ] α i f i /q ≥ ⌈ a/q ⌉ can be combined with P i ∈ [ m ] α i f i ≤ a to derive ≥ .(ii) Suppose that a − P i ∈ I α i b i ≡ b − P i ∈ I α i b i ≡ q . Then ≥ is derived by summing P i ∈ I α i f i ≥ a , P i ∈ I α i f i ≥ b and P i ∈ I α i f i ≤ k I , all of which have already been deduced. † Note that instead of querying the value of P i ∈ I α i f i we could have queried P i ∈ I α i f i ( mod q ) to decrease the numberof leaves to qmd . | I | = 1 then we deduced that a I z = k I for k I b I mod q and we would like to derivea contradiction using the axioms encoding a I z ≡ b I . These axioms are presented to SP as thelinear-inequality encoding of a CNF formula, and while there are no integer solutions satisfyingboth these axioms and a I z = k I , there could in fact be rational solutions. To handle this, wesimply force that each of the at most d variables in a I z takes an integer value by querying thevalue of each variable one by one. As there are at most d variables, each taking an integer valuebetween and q − , this can be done in a tree with at most q d many leaves. At each leaf of thistree we deduce ≥ by a non-negative linear combination with the axioms, the integer-valuedvariables, and a I z ≡ b I .The recursion terminates in at most O (log m ) many rounds because the number of equations underconsideration halves every time. Therefore, the size of this refutation is ( qmd ) O (log m ) q d . Note that bymaking each query in a balanced tree, this refutation can be carried out in depth O (log ( mqd )) .Finally, we conclude Theorem 1.3. Proof of Theorem 1.3.
Observe that the SP refutation from Theorem 4.1 is facelike. Indeed, to performstep 2 we query ( P i ∈ I α i f i ≤ t − , P i ∈ I α i f i ≥ t ) from t = 1 , . . . , q md . For t = 1 , the halfspace P i ∈ I α i f i ≥ is valid for the current polytope because the polytope belongs to the [0 , n cube. Foreach subsequent query, P i ∈ I α i f i ≥ t − is valid because the previous query deduced P i ∈ I α i f i ≥ t − . Similar arguments show that the remaining queries are also facelike. Thus, Lemma 3.6 completesthe proof.We note that the CP refutations that result from Theorem 1.3 have a very particular structure: theyare extremely long and narrow. Indeed, they has depth n O (log m ) . We give a rough sketch of theargument: it is enough to show that most lines L i in the CP refutation are derived using some previousline L j with j = O ( i ) . This is because the final line would have depth proportional to the size of theproof. To see that the CP refutation satisfies this property, observe that for each node visited in thein-order traversal, the nodes in the right subproof π R depend on the halfspace labelling the root, whichin turn depends on the left subproof π L . CP Refutations
Our results from Section 3 suggest an interesting interplay between depth and size of Cutting Planesproofs. In particular, we note that there is a trivial depth n and exponential size refutation of anyunsatisfiable CNF formula in Cutting Planes; however, it is easy to see that the Dadush–Tiwari proofsand our own quasipolynomial size CP proofs of Tseitin are also extremely deep (in particular, they are superlinear ). Even in the stronger Semantic CP it is not clear that the depth of these proofs can bedecreased. However, this does not hold for SP , which has quasi-polynomial size and poly-logarithmicdepth refutations. This motivates Conjecture 1.6, regarding the existence of a “supercritical” trade-offbetween size and depth for Cutting Planes [11, 50]. The Tseitin formulas are a natural candidate forresolving this conjecture.In this section we develop a new method for proving depth lower bounds which we believe shouldbe more useful for resolving this conjecture. Our method works not only for CP but also for semantic CP . Using our technique, we establish the first linear lower bounds on the depth of Semantic CP refutations of the Tseitin formulas. 14ower bounds on the depth of syntactic CP refutations of Tseitin formulas were established byBuresh-Openheim et al. [15] using a rank-based argument. Our proof is inspired by their work, andso we describe it next. Briefly, their proof proceeds by considering a sequence of polytopes P (0) ⊇ . . . ⊇ P ( d ) where P ( i ) is the polytope defined by all inequalities that can be derived in depth i fromthe axioms in F . The goal is to show that P ( d ) is not empty. To do so, they show that a point p ∈ P ( i ) is also in P ( i +1) if for every coordinate j such that < p j < , there exists points p ( j, , p ( j, ∈ P ( i ) such that p ( j,b ) k = b if k = j and p ( j,b ) k = p k otherwise. The proof of this fact is syntactic: it relies onthe careful analysis of the precise rules of CP .When dealing with Semantic CP , we can no longer analyze a finite set of syntactic rules. Further-more, it is not difficult to see that the aforementioned criterion for membership in P ( i +1) is no longersufficient for Semantic CP . We develop an analogous criterion for Semantic CP given later in thissection. As well, we note that the definition of P ( i ) is not well-suited to studying the depth of bounded-size CP proofs like those in Conjecture 1.6 — there does not appear to be a useful way to limit P ( i ) tobe a polytope derived by a bounded number of halfspaces. Therefore we develop our criterion in thelanguage of lifting, which is more amenable to supercritical tradeoffs [11, 50].Through this section we will work with the following top-down definition of Semantic CP . Definition 5.1.
Let F be an n -variate unsatisfiable CNF formula. An sCP refutation of F is a directedacyclic graph of fan-out ≤ where each node v is labelled with a halfspace H v ⊆ R n (understood as aset of points satisfying a linear inequality) satisfying the following:1. Root.
There is a unique source node r labelled with the halfspace H v = R n (corresponding tothe trivially true inequality ≥ ).2. Internal-Nodes.
For each non-leaf node u with children v, w , we have H u ∩ { , } n ⊆ H v ∪ H w . Leaves.
Each sink node u is labeled with a unique clause C ∈ F such that H v ∩ { , } n ⊆ C − (0) .The above definition is obtained by taking a (standard) sCP proof and reversing all inequalities :now, a line is associated with the set of assignments falsified at that line, instead of the assignments satisfying the line.To prove the lower bound we will need to find a long path in the proof. To find this path we will betaking a root-to-leaf walk down the proof while constructing a partial restriction ρ ∈ { , , ∗} n on thevariables. For a partial restriction ρ , denote by free( ρ ) := ρ − ( ∗ ) and fix( ρ ) := [ n ] \ free( ρ ) . Let the restriction of H by ρ be the halfspace H ↾ ρ := { x ∈ R free( ρ ) : ∃ α ∈ H, α fix( ρ ) = ρ fix( ρ ) , α free( ρ ) = x } . It is important to note that H ↾ ρ is itself a halfspace on the free coordinates of ρ .One of our key invariants needed in the proof is the following. Definition 5.2.
A halfspace H ⊆ R n is good if it contains the all- vector, that is, ( ) n = ( , , . . . , ) ∈ H . We will need two technical lemmas to prove the lower bounds. The first lemma shows that if agood halfspace H has its boolean points covered by halfspaces H , H , then one of the two coveringhalfspaces is also good modulo restricting a small set of coordinates.15 emma 5.3. Let H ⊆ R n be any good halfspace, and suppose H ∩ { , } n ⊆ H ∪ H for halfspaces H , H . Then there is a restriction ρ and an i = 1 , such that | fix( ρ ) | ≤ and H i ↾ ρ is good. The second lemma shows that good halfspaces are robust , in the sense that we can restrict a goodhalfspace to another good halfspace while also satisfying any mod-2 equation.
Lemma 5.4.
Let n ≥ and H ⊆ R n be a good halfspace. For any I ⊆ [ n ] with | I | ≥ and b ∈ { , } ,there is a partial restriction ρ ∈ { , , ∗} n with fix( ρ ) = I such that • M i ∈ I ρ ( x i ) = b and • H ↾ ρ ⊆ R free( ρ ) is good. With these two lemmas one can already get an idea of how to construct a long path in the proof.Suppose we start at the root of the proof; the halfspace is ≥ (which is clearly good) and therestriction we maintain is ρ = ∗ n . We can use the first lemma to move from the current good halfspaceto a good child halfspace while increasing the number of fixed coordinates by at most . However, wehave no control over the two coordinates which are fixed by this move, and so we may fall in danger offalsifying an initial constraint. Roughly speaking, we will use the second lemma to satisfy constraintsthat are in danger of being falsified.We delay the proofs of these technical lemmas to the end of the section, and first see how to provethe depth lower bounds. CP Depth
As a warm-up, we show how to lift lower bounds on Resolution depth to Semantic CP depth by com-posing with a constant-width XOR gadget. If F is a CNF formula then we can create a new formula byreplacing each variable z i with an XOR of new variables x i, , . . . , x i, : z i := XOR ( x i, , . . . , x i, ) = x i, ⊕ · · · ⊕ x i, . We call z i the unlifted variable associated with the output of the XOR gadget applied to the i -th block of variables. Formally, let XOR n : { , } n → { , } n be the application of XOR to each -bit blockof a n -bit string. Let F ◦ XOR n denote the formula obtained by performing this substitution on F andtransforming the result into a CNF formula in the obvious way.The main result of this section is the following. Theorem 5.5.
For any unsatisfiable CNF formula F , depth sCP ( F ◦ XOR n ) ≥
12 depth
Res ( F ) . Key to our lower bound will be the following characterization of Resolution depth by
Prover-Adversary games.
Definition 5.6.
The
Prover–Adversary game associated with an n -variate formula F is played betweentwo competing players, Prover and Adversary. The game proceeds in rounds, where in each round thestate of the game is recorded by a partial assignment ρ ∈ { , , ∗} n to the variables of F .16nitially the state is the empty assignment ρ = ∗ n . Then, in each round, the Prover chooses an i ∈ [ n ] with ρ i = ∗ , and the Adversary chooses b ∈ { , } . The state is updated by ρ i ← b and playcontinues. The game ends when the state ρ falsifies an axiom of F .It is known [49] that depth Res ( F ) is exactly the smallest d for which there is a Prover strategy thatends the game in d rounds, regardless of the strategy for the Adversary.The proof of Theorem 5.5 will follow by using an optimal Adversary strategy for F to construct along path in the Semantic CP proof of F ◦ XOR n . Crucially, we need to understand how halfspaces H transform under XOR n : XOR n ( H ) := { z ∈ { , } n : ∃ x ∈ H ∩ { , } n , XOR n ( x ) = z } . As we have already stated, we will maintain a partial assignment ρ ∈ { , , ∗} n on the n lifted variables. However, in order to use the Adversary, we will need to convert ρ to a partial assignmenton the n unlifted variables. To perform this conversion, for any ρ ∈ { , , ∗} n define XOR n ( ρ ) ∈{ , , ∗} n as follows: for each block i ∈ [ n ] , define XOR n ( ρ ) i = ( XOR ( ρ ( x i, ) , . . . , ρ ( x i, )) if ( i, j ) ∈ fix( ρ ) for j ∈ [4] , ∗ otherwise . We are now ready to prove Theorem 5.5. Fix any Semantic CP refutation of F ◦ XOR n , and supposethat there is a strategy for the Adversary in the Prover-Adversary game of F certifying that F requiresdepth d . Throughout the walk, we maintain a partial restriction ρ ∈ { , , ∗} n to the lifted variablessatisfying the following three invariants with respect to the current visited halfspace H .– Block Closed . In every block either all variables in the block are fixed or all variables in the blockare free.–
Good Halfspace . H ↾ ρ is good.– Strategy Consistent . The unlifted assignment
XOR n ( ρ ) does not falsify any clause in F .Initially, we set ρ = ∗ n and the initial halfspace is ≥ , so the pair ( H, ρ ) trivially satisfy theinvariants. Suppose we have reached the halfspace H in our walk and ρ is a restriction satisfying theinvariants. We claim that H cannot be a leaf. To see this, suppose that H is a leaf, then by definition H ∩ { , } n ⊆ C − (0) for some clause C ∈ F ◦ XOR n . By the definition of the lifted formula, thisimplies that XOR n ( H ) ⊆ D − (0) for some clause D ∈ F . Since ( H, ρ ) satisfy the invariants, thelifted assignment XOR n ( ρ ) does not falsify D , and so by the block-closed property it follows that theremust be a variable z i ∈ D such that all lifted variables in the block i are free under ρ . But then applyingLemma 5.4 to the block of variables { x i, , x i, , x i, , x i, } , we can extend ρ to a partial assignment ρ ′ such that z i = XOR ( ρ ( x i, ) , ρ ( x i, ) , ρ ( x i, ) , ρ ( x i, )) satisfies D . But H ↾ ρ ′ is a projection of H ↾ ρ and so this contradicts that XOR n ( H ) violates D .It remains to show how to take a step down the proof. Suppose that we have taken t < d/ stepsdown the Semantic CP proof, the current node is labelled with a halfspace H , and the partial assignment ρ satisfies the invariants. If H has only a single child H , then H ∩ { , } n ⊆ H ∩ { , } n and ρ will still satisfy the invariants for H . Otherwise, if H has two children H and H then applyingLemma 5.3 to the halfspaces H ↾ ρ, H ↾ ρ, H ↾ ρ we can find an i ∈ { , } and a restriction τ suchthat H i ↾ ( ρτ ) is good and τ restricts at most extra coordinates. Let i , i ∈ [ n ] be the two blocks ofvariables in which τ restricts variables, and note that it could be that i = i .Finally, we must restore our invariants. We do this in the following three step process.17 Query the Adversary strategy at the state XOR n ( ρ ) on variables z i , z i and let b , b ∈ { , } bethe responses.• For i = i , i let I i be the set of variables free in the block i , and note that | I i | ≥ . ApplyLemma 5.4 to H ↾ ( ρτ ) and I i to get new restrictions ρ i , ρ i so that blocks i and i both takevalues consistent with the Adversary responses b , b .• Update ρ ← ρτ ρ i ρ i .By Lemma 5.4 the new restriction ρ satisfies the block-closed and the good halfspace invariants. Ateach step we fix at most two blocks of variables, and thus the final invariant is satisfied as long as t < d/ . This completes the proof. CP Depth Lower Bounds for Unlifted Formulas
Next we show how to prove depth lower bounds directly on unlifted families of F -linear equations.The strength of these lower bounds will depend directly on the expansion of the underlying constraint-variable graph of F .Throughout this section, let F denote a set of F -linear equations. In a Semantic CP proof, wemust encode F as a CNF formula, but while proving the lower bound we will instead work with theunderlying system of equations. For a set F of F -linear equations let G ( F ) := ( F ∪ V, E ) be thebipartite constraint-variable graph defined as follows. Each vertex in F corresponds to an equationin F and each vertex in V correspond to variables x i . There is an edge ( C i , x j ) ∈ E if x j occursin the equation C i . For a subset of vertices X ⊆ F ∪ V define the neighbourhood of X in G F as Γ( X ) := { v ∈ F ∪ V : ∃ u ∈ X, ( u, v ) ∈ E } . Definition 5.7.
For a bipartite graph G = ( U ∪ V, E ) the boundary of a set W ⊆ U is δ ( W ) := { v ∈ V : | Γ( v ) ∩ W | = 1 } . The boundary expansion of a set W ⊆ U is | δ ( W ) | / | W | . The graph G is a ( r, s ) - boundary expander if the boundary expansion of every set W ⊆ U with | W | ≤ r has boundary expansion at least s .If F is a system of linear equations then we say that F is an ( r, s ) -boundary expander if its con-straint graph G F is. The main result of this section is the following theorem, analogous to Theorem 5.5. Theorem 5.8.
For any system of F -linear equations F that is an ( r, s + 3) -boundary expander, depth sCP ( F ) ≥ rs/ . The proof of this theorem follows the proof of Theorem 5.5 with some small changes. As before,we will maintain a partial assignment ρ ∈ { , , ∗} n that will guide us on a root-to-leaf walk througha given Semantic CP proof; we also require that each halfspace H that we visit is good relative to ourrestriction ρ . Now our invariants are (somewhat) simpler: we will only require that F ↾ ρ is a sufficientlygood boundary expander.We first prove an auxiliary lemma that will play the role of Lemma 5.4 in the proof of Theorem 5.8.We note that it follows immediately from Lemma 5.4 and boundary expansion. Lemma 5.9.
Suppose F is a system of F -linear equations that is an ( r, s ) -boundary expander for s > , and suppose F ′ ⊆ F with | F ′ | ≤ r . Let H be a good halfspace. Then there exists a ρ ∈ { , , ∗} n with fix( ρ ) = Γ( F ′ ) such that F ′ is satisfied by ρ , and • H ↾ ρ is good.Proof. We first use expansion to find, for each constraint C i ∈ F ′ , a pair of variables y i, , y i, that arein C i ’s boundary. To do this, first observe that | δ ( F ) ′ | ≥ s | F ′ | > | F ′ | by the definition of boundaryexpansion. The pigeonhole principle then immediately implies that there are variables y i, , y i, ∈ δ ( F ′ ) and a constraint C i ∈ F ′ such that y i, , y i, ∈ C i . Since y i, , y i, do not occur in F ′ \ { C i } , it followsthat F ′ \ { C i } is still an ( r, s ) -boundary expander. So, we update F ′ = F ′ \ { C i } and repeat the aboveprocess.When the process terminates, we have for each constraint C i ∈ F ′ a pair of variables y i, , y i, thatoccur only in C i . Write the halfspace H = P i w i x i ≥ c , and let I = Γ( F ′ ) \ S i ∈ I { y i, , y i, } bethe set of variables occurring in F ′ that were not collected by the above process. We define a partialrestriction ρ with fix( ρ ) = I that depends on | I | as follows.• If | I | = 0 then ρ = ∗ n .• If I = { x i } then define ρ ( x i ) = 1 if w i ≥ and ρ ( x i ) = 0 otherwise, and for all other variablesset ρ ( x ) = ∗ .• If | I | > then apply Lemma 5.4 to generate a partial restriction ρ with fix( ρ ) = I that sets the XOR of I arbitrarily.Observe that H ↾ ρ is good. The only non-trivial case is when | I | = 1 , but, in this case we observe ( H ↾ ρ )((1 / n − ) = w i ρ ( x i ) + X j = i w i / ≥ X i w i / ≥ c, where we have used that H is good and the definition of ρ .Next we extend ρ as follows: for each i = 1 , , . . . , | F ′ | apply Lemma 5.4 to I i = { y i, , y i, } togenerate a partial restriction ρ i with fix( ρ i ) = I i so that the constraint C i ↾ ρρ · · · ρ i − is satisfiedby ρ i . Observe that this is always possible since I i is in the boundary of C i . Finally, we update ρ ← ρρ · · · ρ | F ′ | . It follows by Lemma 5.4 that F ′ is satisfied by ρ and H ↾ ρ is good.We are now ready to prove Theorem 5.8. Fix any Semantic CP refutation of F and let n be thenumber of variables. We take a root-to-leaf walk through the refutation while maintaining a partialassignment ρ ∈ { , , ∗} n and an integer valued parameter k ≥ . Throughout the walk we maintainthe following invariants with respect to the current halfspace H :– Good Expansion. F ↾ ρ is a ( k, t ) -boundary expander with t > .– Good Halfspace. H ↾ ρ is good.– Consistency.
The partial assignment ρ does not falsify any clause of F .Initially, we set k = r and ρ = ∗ n , so the invariants are clearly satisfied since F is an ( r, s + 3) -expander. So, suppose that we have reached a halfspace H in our walk, and let k, ρ be parameterssatisfying the invariants. We first observe that if k > then H cannot be a sink node of the proof.To see this, it is enough to show that H contains a satisfying assignment for each equation C ∈ F .Because H ↾ ρ is non-empty (since it is good) there exists a satisfying assignment in H for every19quation satisfied by ρ , so, assume that C is not satisfied by ρ . In this case, since F ↾ ρ is a ( k, t ) -expander for k > we can apply Lemma 5.9 to { C } and H ↾ ρ and obtain a partial restriction τ with fix( τ ) = Γ( C ) such that τ satisfies C . It follows that H is not a leaf.Next, we show how to take a step down the proof while maintaining the invariants. If H has onlya single child H , then H ⊆ H and we can move to H without changing ρ or k . Otherwise, let thechildren of H be H and H . Applying Lemma 5.3 to H ↾ ρ, H ↾ ρ, H ↾ ρ we get a partial restriction τ and an i ∈ { , } such that H i ↾ ρτ is good and | fix( τ ) | ≤ . Due to this latter fact, since F ↾ ρ isa ( k, t ) -expander it follows that F ↾ ρτ is a ( k, t − -expander in the worst case. Observe that since t > it follows that F ↾ ρτ still satisfies the consistency invariant. It remains to restore the expansioninvariant.To restore the expansion invariant, let W be the largest subset of equations such that W has bound-ary expansion at most in F ↾ ρτ , and note that | W | ≤ k and W has boundary expansion at least t − > . Applying Lemma 5.9, we can find a restriction ρ ′ such that W ↾ ρτ ρ ′ is satisfied and H ↾ ρτ ρ ′ is a good halfspace. Now update ρ ← ρτ ρ ′ and k ← k − | W | . Since W is the largest subsetwith expansion at most , it follows that F ↾ ρ is now a ( k, t ′ ) -boundary expander with t ′ > . Finally,we halt the walk if η = 0 .We now argue that this path must have had depth at least rs/ upon halting. Assume that we havetaken t steps down the proof. For each step i ≤ t let W i be the set of equations which lost boundaryexpansion during the i th cleanup step. Note that W i ∩ W j = ∅ for every i = j . Let W ∗ = ∪ ti =1 W i ,note that | W ∗ | = d , and at the end of the walk, W ∗ has no neighbours and therefore no boundary in F ↾ ρ . Before the start of the i th cleanup step, W i has at most | W i | boundary variables. Therefore,at most | W ∗ | = 3 r boundary variables were removed during the cleanup step. Since F started as an ( r, s + 3) -boundary expander, it follows that W had at least r ( s + 3) boundary variables at the start ofthe walk. But, since all variables have been removed from the boundary by the end, this means that rs variables must have been removed from the boundary during the move step. Thus, as each move stepsets at most variables, it follows that t ≥ rs/ before the process halted. In this section we prove our two key technical lemmas: Lemma 5.3 and Lemma 5.4. We begin byproving Lemma 5.4 as it is simpler.
Proof of Lemma 5.4.
Let H be represented by P i ∈ [ n ] w i x i ≥ c and suppose without loss of generalitythat c ≥ and that I = { , . . . , k } . Let the weights of I in H be ordered | w | ≥ | w | ≥ . . . | w k | .Define ρ by setting ρ ( x i ) = ∗ for i I , for i ≤ k − set ρ ( x i ) = 1 if w i ≥ and ρ ( x i ) = 0 otherwise,and set ρ ( x k ) so that L i ∈ I ρ ( x i ) = b . Clearly the parity constraint is satisfied, we show that H ↾ ρ isgood. This follows by an easy calculation: ( H ↾ ρ )((1 / [ n ] \ I ) = w k − ρ ( x k − ) + w k ρ ( x k ) + X i ≤ k − w i ρ ( x i ) + X i ≥ k +1 w i / ≥ w k − / w k / X i ≤ k − w i ρ ( x i ) + X i ≥ k +1 w i / ≥ X i ∈ [ n ] w i / ≥ c where the first inequality follows by averaging since | w k − | ≥ | w k | , and the final inequality followssince H is good. 20n the remainder of the section we prove Lemma 5.3. It will be convenient to work over {− , } n rather than { , } n , so, we restate it over this set and note that we can move between these basis byusing the bijection v (1 − v ) / . Lemma 5.10.
Let H ∈ R n be a halfspace such that n ∈ H and suppose that H ∩{− , } n ⊆ H ∪ H .Then one of H or H contains a point y ∈ {− , , } n such that y has at most two coordinates in {− , } . The key ingredient in our proof of Lemma 5.10 is the following simple topological lemma, whichwill allow us to find a well-behaved point lying on a -face of the {− , } n cube Definition 5.11 ( -face) . A -face of the n -cube with vertices {− , } n are the -dimensional -by- squares spanned by four vertices of the cube that agree on all but two coordinates. Lemma 5.12.
Let w (1) , w (2) ∈ R n be any pair of non-zero vectors, then we can find a vector v ∈ R n orthogonal to w (1) , w (2) , such that v lies on a -face.Proof. We will construct the vector v iteratively by rounding one coordinate at a time until v contains n − coordinates in {− , } . At each step, we will maintain that v ∈ [ − , n , and therefore when theprocess halts v will lie on a -face. For the base case, let u any non-zero vector that is orthogonal to w (1) and w (2) . Begin at the origin by letting v := 0 n and move in the direction of u until one of thecoordinates of v ′ becomes +1 or − . More formally, let α > be the minimum value such that v + αu has one of its coordinates in {− , } . Let j ∈ { , } , and observe that by orthogonality, h v + αu, w ( j ) i = h v, w ( j ) i + α h u, w ( j ) i = h , w ( j ) i + 0 = 0 . Set v to be v + αu . Note that the v are in [ − , , as otherwise this would contradict our choice of α .Suppose that we have constructed a vector v orthogonal to w (1) and w (2) such that all of its coor-dinates belong to [ − , and exactly i < n − of its coordinates belong to {− , } ; suppose w.l.o.g.that they are the first i coordinates. We will show how to booleanize at least one more coordinate of v .Let u be any non-zero vector orthogonal to { w (1) , w (2) , e , . . . , e i } where e j is the j th standard basisvector. Begin moving from v in the direction of u , and let α > be the smallest value such that one ofthe coordinates j > i of v + αu is in {− , } . We verify that the following properties hold:1. The first i coordinates of v + αu are in {− , } . This follows because we moved in a directionorthogonal to e , . . . , e i .2. v + αu is orthogonal to w (1) and w (2) . Let j ∈ { , } and observe that h v i +1 , w ( j ) i = h v i + αu, w ( j ) i = h v i , w ( j ) i + α h u, w ( j ) i = 0 , where the last equality follows because w ( j ) is orthog-onal to v i by induction and u by assumption.Finally, set v to be v + αu . Proof of Lemma 5.10.
Let each of the halfspaces H i be represented by inequalities h w ( i ) x i ≥ b ( i ) for i ∈ { , } . By the previous lemma we can find a point p that lies on a -face F such that p is orthogonalto w (1) and w (2) . Now, p falls into one of the four -by- quadrants of the -face (see the Figure 3). Sup-pose by symmetry that p is contained within the quadrant with vertices { ( − , , (0 , , ( − , , (0 , } ,and let a ∈ F ⊆ R n be the point corresponding to the vertex ( − , .21 a − p (0 ,
0) (1 , , − a = ( − , − , − Figure 3: A -face of the n -cube together with a depiction of the booleanizing process.Since H contains the origin, H contains either v or − v for every vector v . Thus, by symmetry,we can assume that a is contained in H — otherwise, simply exchange a and p for − a and − p . Since H ∩ {− , } n ⊆ H ∪ H and a ∈ {− , } n , it follows that a is in one of H or H . Assume that a ∈ H , so, h w (1) , a i ≥ b (1) . Our goal is to construct a y ∈ H that satisfies the statement of thelemma. We consider two cases:– Case 1: h w (1) , a − p i ≤ . In this case y := 0 n ∈ H as h w (1) , n i = h w (1) , p i ≥ h w (1) , a i ≥ b (1) . where first equality follows because w (1) and p are orthogonal by assumption, and the finalinequality follows because a ∈ H .– Case 2: h w (1) , a − p i > . For simplicity, suppose that the unfixed coordinates of the -face F are { , } . We construct a point that satisfies the statement of the lemma as follows. First, notethat since a, p ∈ F it follows that the vector a − p has at most non-zero coordinates. Beginningat the origin n , move in the direction a − p until a coordinate i ∈ { , } becomes fixed to − or , and let α ( a − p ) be the corresponding point. Since ( a − p ) is in the -face, || a − p || ∞ ≤ and so α ≥ . We can then verify that α ( a − p ) ∈ H , since h w (1) , α ( a − p ) i = α h w (1) , a i ≥ h w (1) , a i ≥ b (1) , where we have used the fact that p is orthogonal to w (1) and α ≥ . Finally, since α ( a − p ) ∈ H we can round the final non-zero coordinate to − or ; since H is a halfspace one of the twovectors will remain in H .In either case we have constructed a vector y ∈ H ( − such that y i ∈ {− , , } n and y i ∈ {− , } forat most two coordinates. We now use the theorems from the previous sections to obtain several concrete lower bounds. First, wegive strong depth lower bounds for sCP proofs of Tseitin formulas on expander graphs.
Theorem 5.13.
There exists a graph G and labelling ℓ : V → { , } such that any sCP refutation of Tseitin ( G, ℓ ) requires depth Ω( n ) . roof. A graph G = ( V, E ) is a γ - edge expander if min {| Γ( W ) | : W ⊆ V, | W | ≤ | V | / } ≥ γ | W | , where Γ( W ) is the neighbourhood of W . We claim that if G is a γ -edge expander then any Tseitinformula over G is a ( n/ , γ ) -boundary expander. Fix any subset W of the equations with | W | ≤ n/ . By the definition of edge expansion we have that | Γ( W ) | ≥ γ | W | , and since each variable iscontained in exactly two constraints, it follows that the boundary of W in Tseitin ( G, ℓ ) has size at least | δ ( W ) | ≥ γ | W | . The result then follows from Theorem 5.8 and the existence of strong edge expanders G (e.g. d -regular Ramanujan graphs are at least d/ -edge expanders, and exist for all d and n [44]).Next, we give lower bounds on the depth of Semantic CP refutations of random k -XOR and random k -CNF formulas for constant k . Definition 5.14.
Let
XOR ( m, n, k ) be the distribution on random k - XOR formulas obtained by sam-pling m equations from the set of all mod 2 linear equations with exactly k variables. Theorem 5.15.
The following holds for Semantic CP :1 . For any k ≥ there exists m = O ( n ) such that F ∼ XOR ( m, n, k ) requires refutations of depthat least Ω( n ) with high probability. . For any k ≥ there exists m = O ( n ) such that F ∼ F ( m, n, k ) requires refutations of depth atleast Ω( n ) with high probability.Proof. We first prove (1) and obtain (2) via a reduction. Fix m = O ( n ) so that F is unsatisfiable withhigh probability. For any constant k, δ and m = O ( n ) , F ∼ XOR ( m, n, k ) is an ( αn, k − − δ ) -boundary expander for some α > (see e.g. [15, 18]). Thus, setting k ≥ and ε to be some smallconstant, the boundary expansion of G F is at least . By Theorem 5.8, F requires depth Ω( n ) to refutein Semantic CP with high probability.The proof of (2) is via a reduction from F ( m, n, k ) to XOR ( m, n, k ) . Every k -clause occurs in theclausal encoding of exactly one k - XOR constraint. It follows that from any k -CNF formula F we cangenerate a k - XOR formula whose clausal expansion F ′ contains F as follows: for each clause C ∈ F ,if C contains an even (odd) number of positive literals then add to F ′ every clause on the variables of C which contains an even (odd) number of positive literals. The resulting F ′ is the clausal encoding ofa set of | F | k - XOR constraints. As there is a unique k - XOR consistent with the clauses of F , we candefine the distribution XOR ( m, n, k ) equivalently as follows:1. Sample F ∼ F ( m, n, k ) ,2. Return the k -XOR F ′ generated from F according to the aforementioned process.It follows that the complexity of refuting F ∼ F ( m, n, k ) is at least that of refuting F ′ ∼ XOR ( m, n, k ) and (2) follows from (1) with the same parameters.Finally, we use Theorem 5.8 to extend the integrality gaps from [15] to sCP by essentially the sameargument. For a linear program with constraints given by a system of linear inequalities Ax ≤ b ,the r -round sCP relaxation adds all inequalities that can be derived from Ax ≤ b by a depth- r sCP proof. We show that the r -round Semantic sCP linear program relaxation cannot well-approximate thenumber of satisfying assignments to a random k -SAT or k -XOR instance.23irst we define our LP relaxations. Suppose that F is a k -CNF formula with m clauses C , C , . . . , C m and n variables x , x , . . . , x n . If C i = W i ∈ P x i ∨ W i ∈ N x i then let E ( C i ) = P i ∈ P x i + P i ∈ N − x i .We consider the following LP relaxation of F : max m X i =1 y i subject to E ( C i ) ≥ y i ∀ i ∈ [ m ]0 ≤ x j ≤ ∀ j ∈ [ n ]0 ≤ y i ≤ ∀ i ∈ [ m ] If F is a k -XOR formula with m constraints and n variables then we consider the above LP relax-ation obtained by writing F as a k -CNF. Finally, recall that the integrality gap is the ratio between theoptimal integral solution to a linear program and the optimal solution produced by the LP. Theorem 5.16.
For any ε > and k ≥ , . There is κ > and m = O ( n ) such that for F ∼ XOR ( m, n, k ) the integrality gap of the κn -round sCP relaxation of F is at least (2 − ε ) with high probability. . There is κ > and m = O ( n ) such that for F ∼ F ( m, n, k ) the integrality gap of the κn -round sCP relaxation of F is at least k / (2 k − − ε with high probability.Proof. Let F ∼ XOR ( m, n, k ) and let Y i be the event that the i th constraint is falsified by a uniformlyrandom assignment. Let δ := ε/ (2 − ε ) , then by a multiplicative Chernoff Bound, the probability that auniformly random assignment satisfies at least a / (2 − ε ) -fraction of F is Pr[ P i ∈ [ m ] Y i ≥ (1+ δ ) m ] ≤ − δm/ . By a union bound, the probability that there exists an assignment satisfying at least a / (2 − ε ) fraction of F is n − δm/ which is exponentially small when m ≥ n (2 − ε ) /ε .On the other hand, consider the partial restriction to the LP relaxation of F that sets y i = 1 for all i ∈ [ m ] . Setting m ≥ n (2 − ε ) /ε large enough, by Theorem 5.15 there some κ > such that withhigh probability F requires depth κn . Hence, the κn round Semantic CP LP relaxation is non-empty,and there is a satisfying assignment α ∈ R n . Thus α ∪ { y i = 1 } satisfies all constraints of max( F ) .The second result follows by an analogous argument. We end by discussing some problems left open by this paper. The most obvious of which is a reso-lution to Conjecture 1.6. Similarly, can supercritical size-depth tradeoffs be established for monotonecircuits? As a first step towards both of these, can one prove a supercritical size-depth tradeoff for aweaker proof system such as Resolution?The simulation results presented in Section 3 leave open several questions regarding the relation-ship between SP and CP . First, the simulation of SP ∗ by CP incurs a significant blowup in the co-efficient size due to Shrijver’s lemma. It would be interesting to understand whether SP ∗ can quasi-polynomially simulated by CP ∗ ; that is, whether this blowup in the size of the coefficients is necessary.The most obvious question left open by these simulations is whether CP can polynomially simulate SP , or even polynomially simulate SP ∗ . Similarly, what are the relationships of both SP and CP , to(bounded-coefficient) R ( CP ) , the system which corresponds to dag-like SP . R ( CP ) can polynomiallysimulate DNF Resolution, and therefore has polynomial size proofs of the Clique-Colouring formulas,24or cliques of size Ω( √ n ) and colourings of size o (log n ) [4]. Quasi-polynomial lower bounds onthe size of CP refutations are known for this range of parameters and this rules out a polynomialsimulation by Cutting Planes; however, a quasi-polynomial simulation may be possible. Extending thispolynomial upper bound to work in the case to closer clique and coloring sizes by taking advantage ofthe additional power of R ( CP ) over DNF Resolution appears to be a promising approach to resolvingthis question. Acknowledgements
T.P. was supported by NSERC, NSF Grant No. CCF-1900460 and the IAS school of mathematics.R.R. was supported by NSERC, the Charles Simonyi Endowment, and indirectly supported by theNational Science Foundation Grant No. CCF-1900460. L.T. was supported by NSF grant CCF-192179and NSF CAREER award CCF-1942123. Any opinions, findings and conclusions or recommendationsexpressed in this material are those of the author(s) and do not necessarily reflect the views of theNational Science Foundation.
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