Depth lower bounds in Stabbing Planes for combinatorial principles
aa r X i v : . [ c s . CC ] F e b Depth lower bounds in Stabbing Planes for combinatorialprinciples
Stefan DantchevDepartment of Computer ScienceDurham University, UK [email protected]
Nicola GalesiDepartment of Computer ScienceSapienza Universit`a di Roma, IT [email protected]
Abdul GhaniDepartment of Computer ScienceDurham University, UK [email protected]
Barnaby MartinDepartment of Computer ScienceDurham University, UK [email protected]
February 16, 2021
Abstract
We prove logarithmic depth lower bounds in
Stabbing Planes for the classes of combinatorialprinciples known as the
Pigeonhole principle and the
Tseitin contradictions. The depth lower boundsare new, obtained by giving almost linear length lower bounds which do not depend on the bit-sizeof the inequalities and in the case of the Pigeonhole principle are tight.The technique known so far to prove depth lower bounds for Stabbing Planes is a generalizationof that used for the
Cutting Planes proof system. In this work we introduce two new approachesto prove length/depth lower bounds in Stabbing Planes: one relying on
Sperner’s Theorem whichworks for the Pigeonhole principle and Tseitin contradictions over the complete graph; a secondproving the lower bound for Tseitin contradictions over a grid graph, which uses a result on essentialcoverings of the boolean cube by linear polynomials, which in turn relies on
Alon’s combinatorialNullenstellensatz . Finding a satisfying assignment for a propositional formula (
SAT ) is a central component for manycomputationally hard problems. Despite being older than 50 years and exponential time in the worst-case, the
DPLL algorithm [6, 7, 17] is the core of essentially all high performance modern
SAT -solvers.
DPLL is a recursive boolean method: at each call one variable x of the formula F is chosen and thesearch recursively branches into the two cases obtained by setting x respectively to and in F . On UNSAT formulas
DPLL performs the worst and it is well-known that the execution trace of the
DPLL algorithm running on an unsatisfiable formula F is nothing more than a treelike refutation of F in theproof system of Resolution [17] (
Res ).Since
SAT can be viewed as an optimization problem the question whether Integer Linear Program-ming (
ILP ) can be made feasible for satisfiability testing received a lot of attention and is consideredamong the most challenging problems in local search [18, 10]. One proof system capturing
ILP ap-proaches to
SAT is Cutting Planes , a system whose main rule implements the rounding (or
Ch´avtal cut )approach to
ILP . Cutting planes works with integer linear inequalities of the form ax ≤ b , with a , b in-tegers, and, like resolution, is a sound and complete refutational proof system for CNF formulas: indeeda clause C = ( x ∨ . . . ∨ x r ∨ ¬ y ∨ . . . ∨ ¬ y s ) can be written as the integer inequality y − x ≤ s − .1eame et al. [2], extended the idea of DPLL to a more general proof strategy based on
ILP . Insteadof branching only on a variable as in resolution, in this method one considers a pair ( a , b ) , with a ∈ Z n and b ∈ Z , and branches limiting the search to the two half-planes: ax ≤ b − and ax ≥ b . A path terminates when the LP defined by the inequalities in F and those forming the path is infeasible. Thismethod can be made into a refutational treelike proof system for UNSAT
CNF’s called
Stabbing planes ( SP ) ([2]) and it turned out that it is polynomially equivalent to the treelike version of Res ( CP ) , a proofsystem introduced by Kraj´ıˇcek [12] where clauses are disjunction of linear inequalities.In this work we consider the complexity of proofs in SP focusing on the length , i.e. the number ofqueries in the proof; the depth (called also rank in [2]), i.e. the length of the longest path in the prooftree; and the size , i.e. the bit size of all the coefficients appearing in the proof. Lower bounds for size can be obtained in SP , but in a limited way: in [2] it is proven that size S anddepth D SP refutations imply treelike Res ( CP ) proofs of size O ( S ) and width O ( D ) ; Kojevnikov [11],improving the interpolation method introduced for Res ( CP ) by Kraj´ıˇcek [12], gave exponential lowerbounds for treelike Res ( CP ) when the width of the clauses (i.e. the number of linear inequalities in aclause) is bounded by o ( n/ log n ) . Hence these lower bounds are applicable only to very specific classesof formulas (whose hardness comes from boolean circuit hardness) and only to SP refutations of lowdepth.Nevertheless SP appears to be a strong proof system. Firstly notice that the condition terminating apath in a proof is not a trivial contradiction like in resolution, but is the infeasibility of an LP , which isonly a polynomial time verifiable condition. Hence linear size SP proofs might be already a strong classof SP proofs, since they can hide a polynomial growth into one final node whence to run the verificationof the terminating condition. At present we know that:1. SP polynomially simulates CP (Theorem 4.5 in [2]). Hence in particular the PHP mn can be refutedin SP by a proof of size O ( n ) ([5]). Furthermore it can be refuted by a O (log n ) depth proofsince polynomial size CP proofs, by Theorem 4.4 in [2], can be balanced in SP .2. Beame et al. in [2] proved the surprising result that the class of Tseitin contradictions Ts ( G, ω ) over any graph G of maximum degree D , with an odd charging ω , can be refuted in SP in sizequasipolynomial in | G | and depth O (log | G | + D ) .Depth lower bounds for SP are proved in [2]:1. a Ω( n/ log n ) lower bound for the formula Ts ( G, w ) ◦ VER n , composing Ts ( G, ω ) (over anexpander graph G ) with the gadget function VER n (see Theorem 5.7 in [2] for details); and2. a Ω( √ n log n ) lower bound for the formula Peb ( G ) ◦ IND nl over n + n log n variables obtainedby lifting a pebbling formula Peb ( G ) over a graph with high pebbling number, with a pointerfunction gadget IND nl (see Theorem 5.5. in [2] for details).Similarly to size, these depth lower bounds are also applicable only to very specific classes of for-mulas. In fact they are obtained by extending to SP the technique introduced by Kraj´ıˇcek [13] for CP ofreducing shallow proofs of a formula F to efficient real communication protocols computing a relatedsearch problem and then proving that such efficient protocols cannot exist.Despite the fact that SP is at least as strong as CP , in SP the known lower bounds techniques arederived from those of treelike CP . Hence finding other techniques to prove depth and size lower bounds Another way of proving this result is using Theorem 4.8 in [2] stating that if there are length L and space S CP refuta-tions of a set of linear integral inequalities, then there are depth O ( S log L ) SP refutations of the same set of linear integralinequalities; and then use the result in [9] (Theorem 5.1) that PHP mn has polynomial length and constant space CP refutations. SP is important to understand its proof strength. For instance, unlike CP where we know tight Θ(log n ) rank bounds for the PHP mn [3, 16] and Ω( n ) rank bounds for Tseitin contradictions [3], for SP no depth lower bound is at present known for purely combinatorial statements.In this work we address such problems. The main original motivation of this work was to prove depth lower bounds in SP for truly combinatorialstatements, like Ts ( G, w ) or PHP mn , which we know to be efficiently provable, but on which we cannotuse methods reducing to the complexity of boolean functions, like the ones mentioned above. We presenttwo new methods for proving depth lower bounds in SP which in fact are the consequence of provinglength lower bounds. Our bounds are numerically weak (almost linear for the length and logarithmic forthe depth), but for the PHP mn they give optimal depth lower bounds from length lower bounds that donot depend on the bit-size of the coefficients. We prove:1. an optimal Ω(log n ) lower bound for the depth of SP proofs of the PHP mn .2. an Ω(log n ) lower bound for the depth of SP proofs of Ts ( G, ω ) , when G is a n × n grid graph H n or the complete graph K n . These last results must be compared with the O (log n ) upper boundfor Ts ( H n , ω ) given in [2].Our results are derived from the following initial geometrical observation: let S be a space of admissible points in { , , / } n satisfying a given unsatisfiable system of integer linear inequalities F ( x , . . . , x n ) . In a SP proof for F , at each branch Q = ( a , b ) the set of points in the slab ( Q ) = { s ∈ S : b − < ax < b } does not survive in S . At the end of the proof on the leaves, where we haveinfeasible LP ’s, no point in S can survive the proof. So it is sufficient to find conditions such that, underthe assumption that a proof of F is “small”, even one point of S survives the proof. In pursuing thisapproach we use two methods.The antichain method . Here we use a well-known bound based on Sperner’s Theorem [4, 20] toupper bound the number of points in the slabs where the set of non-zero coefficients is sufficiently large.Trading between the number of such slabs and the number of points ruled out from the space S ofadmissible points, we obtain the lower bound.We initially present the method and the Ω(log n ) lower bound on a set of unsatisfiable integer linearinequalities - the Simple Pigeonhole Principle ( SPHP ) - capturing the core of the counting argumentused to prove efficiently the
PHP in CP . Since SPHP n has rank CP proofs, it entails a strong separationbetween CP rank and SP depth. We then apply the method to PHP mn and to Ts ( K n , ω ) .The covering method . The antichain method appears too weak to prove size and depth lower boundson Ts ( G, w ) , when G is for example a grid or a pyramid. To solve this case, we consider anotherapproach that we call the covering method : we reduce the problem of proving that one point in S survivesfrom all the slab ( Q ) in a small proof of F , to the problem that a set of polynomials which essentiallycovers the boolean cube { , } n requires at least n polynomials, which is a well-known problem [1, 14].For this reduction to work we have to find a high dimensional projection of S covering the boolean cubeand defined on variables effectively appearing in the proof. We prove that matchings in G work properlyto this aim on Ts ( G, ω ) . Since the grid H n has large matchings, we can obtain the lower bound on Ts ( H n , ω ) .The paper is organized as follows: We give the preliminary definitions in the next section and thenwe move to other sections: one on the lower bounds by the antichain method and the other on lowerbounds by the covering method. 3 Preliminaries
We use [ n ] for the set { , , . . . , n } , Z / for Z ∪ ( Z + ) and Z + for { , , . . . } . Here we recall the definition of the Stabbing Planes proof system from [2].
Definition 1. A linear integer inequality in the variables x , . . . , x n is something of the form P ni =1 a i x i ≥ b , where each a i and b are integral. A set of such inequalities is said to be unsatisfiable if there are no / assignments to the x variables satisfying each inequality simultaneously. Note that we reserve the term infeasible, in contrast to unsatisfiable, for (real or rational) linear programs.
Definition 2.
Fix some variables x , . . . , x n . A Stabbing Planes ( SP ) proof of a set of integer linearinequalities F is a binary tree T , with each node labeled with a query ( a , b ) with a ∈ Z n , b ∈ Z .Out of each node we have an edge labeled with a x ≥ b and the other labeled with its integer negation a x ≤ b − . Each leaf ℓ is labeled with a LP system P ℓ made by a nonnegative linear combination ofinequalities from F and the inequalities labelling the edges on the path from the root of T to the leaf ℓ .If F is an unsatisfiable set of integer linear inequalities, T is a Stabbing Planes ( SP ) refutation of F if all the LP ’s P ℓ on the leaves of T are infeasible. Definition 3.
The slab corresponding to a query Q = ( a , b ) is the set slab ( Q ) = { x ∈ R n : b − < ax < b } satisfying neither of the associated inequalities. Since each leaf in a SP refutation is labelled by an infeasible LP , throughout this paper we willactually use the following geometric observation on SP proofs T : the set of points in R n must all beruled out by a query somewhere in T . In particular this will be true for those points in R n which satisfya set of integer linear inequalities F and which we call feasible points for F . Fact 1.
The slabs associated with a SP refutation must cover the feasible points of F . That is, { y ∈ R n : ay ≥ b for all ( a , b ) ∈ F } ⊆ [ ( a ,b ) ∈F { x ∈ R n : b − < ax < b } The length of a SP refutation is the number of queries in the proof tree. The depth of a SP refutation T is the longest root-to-leaf path in T . The size (respectively depth) of refuting F in SP is the minimum size (respectively depth) over all SP refutations of F . We call bit-size of a SP refutation T the totalnumber of bits needed to represent every inequality in the refutation. Definition 4 ([5]) . The
Cutting Planes (CP) proof system is equipped with boolean axioms and twoinference rules: Boolean Axioms Linear Combination Rounding x ≥ − x ≥− ax ≥ c bx ≥ dα ax + β bx ≥ αc + βd α ax ≥ b ax ≥⌈ b/α ⌉ where α, β, b ∈ Z + and a , b ∈ Z n . A CP refutation of some unsatisfiable set of integer linear inequali-ties is a derivation of ≥ by the aforementioned inference rules from the inequalities in F . A CP refutation is treelike if the directed acyclic graph underlying the proof is a tree. The length ofa CP refutation is the number of inequalities in the sequence. The depth is the length of the longest pathfrom the root to a leaf (sink) in the graph. The rank of a CP proof is the maximal number of roundingrules used in a path of the proof graph. The size of a CP refutation is the bit-size to represent all theinequalities in the proof. 4 .2 Restrictions Let V = { x , . . . , x n } be a set of n variables and let ax ≤ b be a linear integer inequality. We say that avariable x i appears in , or is mentioned by a query Q = ( a , b ) if a i = 0 and does not appear otherwise.A restriction ρ is a function ρ : D → { , } , D ⊆ V . A restriction acts on a half-plane ax ≤ b setting the x i ’s according to ρ . Notice that the variables x i ∈ D do not appear in the restricted half-plane.By T ↾ ρ we mean to apply the restriction ρ to all the queries in a SP proof T . The tree T ↾ ρ definesa new SP proof: if some Q ↾ ρ reduces to ≤ − b , for some b ≥ , then that node becomes a leaf in T ↾ ρ . Otherwise in T ↾ ρ we simply branch on Q ↾ ρ . Of course the solution space defined by the linearinequalities labelling a path in T ↾ ρ is a subset of the solution space defined by the corresponding path in T . Hence the leaves of T ↾ ρ define an infeasible LP .We work with linear integer inequalities which are a translation of families of CNFs F . Hencewhen we write F ↾ ρ we mean the applications of the restriction ρ to the set of linear integer inequalitiesdefining F . This method is based on Sperner’s theorem. Using it we can prove depth lower bounds in SP for PHP mn and for Tseitin contradictions Ts ( K n , ω ) over the complete graph. To motivate and explain the maindefinitions, we use as an example a simplification of the PHP mn , the simplified Pigeonhole principle SPHP n , which has some interest since, as we will show it exponentially separates CP rank from SP depth. As mentioned in the Introduction, the
SPHP n intends to capture the core of the counting argument usedto prove efficiently the PHP in CP . Definition 5.
The
SPHP n is the following unsatisfiable family of inequalities: P ni =1 x i ≥ x i + x j ≤ (for all i = j ∈ [ n ] ) Lemma 1.
SPHP n has a rank CP refutation, for n ≥ .Proof. Let S := P ni =1 x i (so we have S ≥ ). We fix some i ∈ [ n ] and sum x i + x j ≤ over all j ∈ [ n ] \ { i } to find S + ( n − x i ≤ n − . We add this to − S ≤ − to get x i ≤ n − n − which becomes x i ≤ after a single cut. We do this for every i and find S ≤ - a contradiction whencombined with the axiom S ≥ .It is easy to see that SPHP n has depth O (log n ) proofs in SP , either by a direct proof or appealingto the polynomial size proofs in CP of the PHP mn ([5]) and then using the Theorem 4.4 in [2] informallystating that “ CP proofs can be balanced in SP ”. Corollary 1.
The
SPHP n has Stabbing Planes refutations of depth O (log n ) . We will prove that this depth is tight. 5 .2 Sperner’s Theorem
Let a ∈ R n . The width w ( a ) of a is the number of non-zero coordinates in a . The width of a query ( a , b ) is w ( a ) .Let n ∈ N . Fix W ⊆ [0 , ∩ Q + of finite size k ≥ and insist that ∈ W . We also insist that,if w m is the maximal element in W , that W is closed under the map x → w m − x . The W ’s we workwith in this paper are { , / } and { , / , } . We disallow (for example) { , / , } as it is not closedunder x → − x . Definition 6. A ( n, W ) -word is an element in W n . We consider the following extension of Sperner’s theorem.
Theorem 1 ([15, 4]) . Fix some t ≥ , t ∈ N . For the pointwise ordering of [ t ] f , any antichain has sizeat most t f q π ( t − f (1 + o (1)) . We will use the simplified bound that any antichain A has size |A| ≤ t f √ f . Lemma 2.
Let a ∈ Z n and | W | = k ≥ . The number of ( n, W ) -words s such that as = b , where b ∈ Q is at most k n √ w ( a ) .Proof. Assume for a moment that the a i ’s are non-negative. Let I a = { i ∈ [ n ] : a i = 0 } . Notice that | I a | = w ( a ) . For a ( n, W ) -word s , let ˜ s = proj ↾ I a ( s ) be the projection of s on the set of coordinates I a . ˜ s is a ( w ( a ) , k ) -word and we claim that the set of such ˜ s forms an antichain on [ k ] w ( a ) . Let J ˜ s = { i ∈ I a | ˜ s i = 0 } . Since as = b , then P i ∈ J ˜ s a i ˜ s i = b . By the non-negativity of the a i ’s itfollows that if J ˜ s ⊂ J ˜ t , then P i ∈ J ˜ s a i ˜ s i < P i ∈ J ˜ t a i ˜ t i . Hence the set of J ˜ s forms an antichain on [ k ] w ( a ) induced from the ˜ s . That is, we have at most k w ( a ) / p w ( a ) such J ˜ s , each the result of projectingat most n − w ( a ) solution ( n, W ) -words.To justify the assumption that all a i ’s are non-negative, we show how to repeatedly replace the pair a , b with a new pair a ′ , b ′ , where w ( a ′ ) = w ( a ) and the number of ( n, W ) -words s such that as = b isthe same as those with a ′ s = b ′ . We do this until a is nonnegative and the upper bound applies.Let w m be the maximum element in W and write as = P ni ∈ a i s i . Say that a < . Then we simplyreplace s with w m − s , and replace a and b with the result of rearranging a ( w m − s ) + n X a i s i = b which is − a s + n X a i s i = b − a w m . So let a ′ = ( − a , a , . . . , a n ) and b ′ = b − a w m . It is important to note that the pairs a , b and a ′ , b ′ have the same number of solutions in W n . This can be seen by mirroring the first coordinate around w m :the vector x = ( x , . . . , x n ) is a solution to P ni ∈ a i x i = b if and only if x ′ = ( x ′ = w m − x , x ′ = x . . . , x ′ n = x n ) is a solution for − a x ′ + n X a i x ′ i = − a ( w m − x ) + n X a i x i = n X a i x i − a w m = b − a w m = b ′ . Although we note that the following proofs can be modified to work without this demand of closure. .3 Large admissibility A ( n, W ) -word s is admissible for an unsatisfiable set of integer linear inequalities F over n variablesif s satisfies all constraints of F . A set of ( n, W ) -words is admissible for F if all its elements areadmissible. A ( F , W ) is the set of all admissible ( n, W ) -words for F .The interesting sets W for an unsatisfiable set of integer linear inequalities F are those such thatalmost all ( n, W ) -words are admissible for F . We will apply our method on sets of integer linear in-equalities which are a translation of unsatisfiable CNF’s generated over a given domain. Typically theseformulas on a size n domain have a number of variables which is not exactly n but a function of n , ν ( n ) ≥ n . Hence for the rest of this section we consider F := {F n } n ∈ N as a family of sets of unsatis-fiable integer linear inequalities, where F n has ν ( n ) ≥ n variables. We call F an unsatisfiable family .Consider then the following definition recalling that we denote k = | W | : Definition 7. F is almost full if |A ( F n , W ) | ≥ k n − o ( k n ) . Notice that, because of the o notation, Definition 7 might be not necessarily true for all n ∈ N , butonly starting from some n F . Definition 8.
Given some almost full family F (over ν ( n ) variables) we let n F be the natural numberwith k ν ( n ) |A ( F n , W ) | ≤ for all n ≥ n F . As an example we prove
SPHP is almost full (notice that in the case of
SPHP n , ν ( n ) = n ). Lemma 3.
SPHP n is almost full.Proof. Fix W = { , / } so that k = | W | = 2 . Let U be the set of all ( n, W ) -words with at least fourcoordinates set to / . U is admissible for SPHP n since inequalities x i + x j ≤ are always satisfiedfor any value in W and inequalities x + . . . + x n ≥ are satisfied by all points in U which alwayscontain four / . By a simple counting argument, in U there are n − n = 2 n − o (2 n ) admissible ( n, W ) -words. Hence the claim. Lemma 4.
Let F = {F n } n ∈ N be an almost full unsatisfiable family, where F n has ν ( n ) variables.Further let T be a SP refutation of F of minimal width ω . If n ≥ n F then |T | = Ω( √ w ) .Proof. We estimate at what rate the slab of the queries in T rule out admissible points in U .Since all the queries in T have width at least w , according to Lemma 2, each query in T rules outat most k ν ( n ) √ w admissible points. By Fact 1 no point survives at the leaves, in particular the admissiblepoints. Then it must be that |T | k ν ( n ) √ w ≥ |A ( F n , W ) | which means |T | · k ν ( n ) |A ( F n , W ) | ≥ √ w We finish by noting that, by the assumption n ≥ n F , and then by Definition 8, we have ≥ k ν ( n ) |A ( F n ,W ) | . We focus on restrictions ρ that after applied on an unsatisfiable family F = {F n } n ∈ N , reduce the set F to another set in the same family. 7 efinition 9. Let F = {F n } n ∈ N be an unsatisfiable family and c a positive constant. F is c -self-reducible if for any set V of variables, with | V | = t < n/c , there is a restriction ρ with domain V ′ ⊇ V ,such that F n ↾ ρ = F n − ct (up to renaming of variables). Let us motivate the definition with an example.
Lemma 5.
SPHP n is -self-reducible.Proof. Whatever set of variables x i , i ∈ I ⊂ [ n ] we consider, it is sufficient to set x i to to fulfillDefinition 9. Theorem 2.
Let F := {F n } n ∈ N be a unsatisfiable set of integer linear inequalities which is almostfull and c -self-reducible. If F n defines a feasible LP whenever n > n F , then for n large enough, theshortest SP proof of F n is of length Ω( √ n ) .Proof. Take any SP proof T refuting F n and fix t = √ n .The proof proceeds by stages i ≥ where T = T . The stages will go on while the invariantproperty (which at stage is true since n > n F and c a positive constant) n − ict > max { n F , n (1 − /c ) } holds.At the stage i we let Σ i = { ( a , b ) ∈ T i : w ( a ) ≤ t } and s i = | Σ i | . If s i ≥ t the claim is triviallyproven. If s i = 0 , then all queries in T i have width at least t and by Lemma 4 (which can be appliedsince n − ict > n F ) the claim is proven (for n large enough).So assume that < s i < t . Each of the queries in Σ i involves at most t nonzero coefficients,hence in total they mention at most s i t ≤ t variables. Extend this set of variables to some V ′ inaccordance with Definition 9 (which can be done since, by the invariant, ict < n/c ). Set all thesevariables according to self-reducibility of F in a restriction ρ i and define T i +1 = T i ↾ ρ i . Note that byDefinition 9 and by that of restriction, T i +1 is a SP refutation of F n − ict and we can go on with the nextstage. (Also note that we do not hit an empty refutation this way, due to the assumption that F n definesa feasible LP.)Assume that the invariant does not hold. If this is because n − ict < n F then, as each iterationdestroys at least one node, |T | ≥ i > n − n F ct ∈ Ω( n / ) . If this is because n − ict < n − n/c , then again for the same reason it holds that |T | ≥ i > nc n / ∈ Ω( n / ) . Using Lemmas 3 and 5 and the previous Theorem we get:
Corollary 2.
The length of any SP refutation of SPHP n is Ω( √ n ) . Hence the minimal depth is Ω(log n ) . Definition 10.
The
Pigeonhole Principle
PHP mn , m > n , is the family of unsatisfiable integer linearinequalities defined over the variables { P i,j : i ∈ [ m ] , j ∈ [ n ] } consisting of the following inequalities: P nj =1 P ij ≥ ∀ j ∈ [ m ] (every pigeon goes into some hole) P ik + P jk ≤ ∀ k ∈ [ n ] , i = j ∈ [ m ] (at most one pigeon enters any given hole)
8e present a lower bound for
PHP mn closely following that for SPHP n , in which we largely ignorethe diversity of different pigeons (which makes the principle rather like SPHP n ).In this subsection we fix W = { , / } , and for the sake of brevity refer to ( n, W ) -words as biwords . Lemma 6.
The
PHP mn is almost full.Proof. We show that there are at least mn − admissible biwords (for sufficiently large n ). For eachpigeon i , there are admissible valuations to holes so that, so long as at least two of these are set to / ,the others may be set to anything in { , / } . This gives at least n − ( n + 1) possibilities. Since thepigeons are independent, we obtain at least (2 n − ( n + 1)) m biwords. Now this is mn (cid:0) − n +12 n (cid:1) m where (cid:0) − n +12 n (cid:1) m ∼ e − ( n +1) m n whence, (cid:0) − n +12 n (cid:1) m ≥ e − ( n +2) m n for sufficiently large n . It followsthere is a constant c so that: mn (cid:18) − n + 12 n (cid:19) m ≥ mn − c ( n +2) m n ≥ mn − for sufficiently large n . Lemma 7.
The
PHP mn is -self-reducible (with respect to the subscript n ).Proof. We are given some set I of variables from PHP mn . Let H := { j : P i,j ∈ I for some i } be theholes mentioned by I . We simply forbid these holes with the restriction setting P i,j to for every pigeon i ∈ [ m ] and every hole j ∈ H . Theorem 3.
The length of any SP refutation of PHP mn is Ω( n / ) .Proof. Note that the all / point is feasible for PHP mn . Then with Lemma 6 and Lemma 7 in hand wemeet all the prerequisites for Theorem 2.By simply noting that a SP refutation is a binary tree, we get the following corollary. Corollary 3.
The SP depth of the
PHP mn is Ω(log n ) . Definition 11.
For a graph G = ( V, E ) along with a charging function ω : V → { , } satisfying P v ∈ V ω ( v ) = 1 mod 2 . The Tseitin contradiction Ts ( G, ω ) is the set of linear inequalities whichtranslate the CNF encoding of X e ∈ Ee ∋ v x e = ω ( v ) mod 2 . for every v ∈ V , where the variables x e range over the edges e ∈ E . In this subsection we consider Ts ( K n , ω ) and ω will always be an odd charging for K n . We let N := (cid:0) n (cid:1) and we fix W = { , / , } , k = 3 and for the sake of brevity refer to ( n, W ) -words as triwords . Wewill abuse slightly the notation of Section 3.3 and consider the family { Ts ( K n , ω ) } n ∈ N , ω odd as a singleparameter family in n . The reason we can do this is because the following proofs of almost fullness andself reducibility do not depend on ω at all (so long as it is odd, which we will always ensure). Lemma 8. Ts ( K n , ω ) is almost full. roof. We show that Ts ( K n , ω ) has at least c N admissible triwords, for any constant < c < and n large enough. We define the assignment ρ setting all edges (i.e. x e ) to a value in W = { , , / } independently and uniformly at random, and inspecting the probability that some fixed constraint for anode v is violated by ρ .Clearly if at least edges incident to v are set to / its constraint is satisfied. If none of its incidentedges are set to / then it is satisfied with probability / . Let A ( v ) be the event “ no edge incident to v is set to / by ρ ” and let B ( v ) be the event that “ exactly one edge incident to v is set to / by ρ ”.Then: Pr[ v is violated ] ≤
12 Pr[ A ( v )] + Pr[ B ( v )] = 12 2 n − n − + ( n − n − n − = n n − n − . Therefore, by a union bound, the probability that there exists a node with violated parity is boundedabove by n n − n − , which approaches as n goes to infinity. Lemma 9. Ts ( K n , ω ) is -self-reducible.Proof. We are given some set of variables I . Each variable mentions nodes, so extend these mentionednodes arbitrarily to a set S of size exactly | I | , which we then hit with the following restriction: if S isevenly charged, pick any matching on the set { s ∈ S : w ( s ) = 1 } , set those edges to , and set any otheredges involving some vertex in S to . Otherwise (if S is oddly charged) pick any l ∈ { s ∈ S : w ( s ) =1 } and r ∈ [ n ] \ S and set x lr to . { s ∈ S : w ( s ) = 1 } \ l is now even so we can pick a matching asbefore. And as before we set all other edges involving some vertex in S to 0. In the first case the graphinduced by [ n ] \ S must be oddly charged (as the original graph was). In the second case this inducedgraph was originally evenly charged, but we changed this when we set x lr to . Lemma 10.
For any oddly charged ω and n large enough, all SP refutations of Ts ( K n , ω ) have length Ω( √ n ) .Proof. We have that the all / point is feasible for Ts ( K n , ω ) . Then we can simply apply Theorem 2. Corollary 4.
The depth of any SP refutation of Ts ( K n , ω ) is Ω(log n ) . In [14] Linial and Radhakrishnan considered the problem of the minimal number of hyperplanes cov-ering all the points of the cube { , } n . To make the problem meaningful they defined the notion of an essential covering of { , } n . Definition 12 ([14]) . A set L of linear polynomials with real coefficients is said to be an essential cover of the cube { , } n if(E1) for each v ∈ { , } n , there is a p ∈ L such that p ( v ) = 0 ,(E2) no proper subset of L satisfies (E1), that is, for every p ∈ L , there is a v ∈ { , } n such that p alone takes the value on v , and(E3) every variable appears (in some monomial with non-zero coefficient) in some polynomial of L . They also prove the following theorem:
Theorem 4 ([14], Theorem 2) . Any essential cover L of the cube with n coordinates satisfies | L | ∈ Ω( √ n ) . .1 Turning a refutation of Tseitin contradictions into an essential cover of the hyper-cube Fix some Ts ( G, ω ) and a SP refutation T , thought of as a set of queries ( a , b ) . We say that an edge of G is mentioned in T if the variable x e appears with non-zero coefficient in some query in T . Lemma 11.
For any matching M on the edges of G mentioned in T , where all matched vertices havedegree at least , there is an essential cover L of the | M | coordinate hypercube with | L | ≤ | R | .Proof. Let M be any matching on the edges mentioned in T . Let H ′ be the set of the | M | admissiblepoints for Ts ( G, ω ) gotten by giving the edges in M any { , } value and setting the rest of the edges to / . (That these are admissible for Ts ( G, ω ) comes from the fact that, given the degree of G is at least , all vertices, including matched ones, are incident to at least two edges set to / .) By Fact 1 all ofthese points must have been killed in some query ( a , b ) in T (i.e. x ∈ slab ( a , b )) ). Hence x is killed by ( a , b ) ⇔ X e ∈ E ( G ) a e x e = b + 1 / ⇔ X e ∈ M a e x e = b + 1 / − (1 / X e ∈ E ( G ) \ M a e This just means that the set L := ( a e ) e ∈ M , b + 12 − X e ∈ E ( G ) \ M a e : ( a, b ) ∈ R covers the hypercube H = { , } | M | defined over variables x e , e ∈ M . It remains to show that thiscover is essential:• (E1) i.e. for each x ∈ H , there is some p ∈ L with p ( x ) = 0 . This is clear and just talked about.• (E2) i.e. no proper subset satisfies E1. As we are interested in the lower bound, we can just takesubsets if we need to.• (E3) i.e. every variable appears with non-zero coefficient somewhere in L . This just follows from M being a subset of the edges mentioned by T . Lemma 12.
Pick some set S of vertex-disjoint unit squares (4-cycles) in H n . Any refutation R of any Ts ( H n , ω ) must mention at least one edge in each square.Proof. We will use a couple times the following idea due to A. Urquhart in [19]: starting from somebinary assignment to the edges in G , take a path from two vertices u and v , and flip all the edges onthis path. If u and v are distinct, we flip their polarities (as they are incident to exactly one edge thatgets flipped) and no other vertex has its incident charge changed (because they have zero or two incidentedges flipped.) If u = v , then nothing gets its polarity changed. In this way we can find a binaryassignment to the edges of any graph falsifying exactly the parity constraint for a single node v .So suppose some square s := { a, b, c, d } ⊆ V ( H n ) fails to have a single edge mentioned. Find abinary assignment to the edges falsifying (say) only the parity constraint for a . Modify this assignmentby setting every edge in s to / . This point is admissible - a is incident to two / edges and so is anyother vertex that has had its incident edges touched. But this admissible point can never be ruled out ina slab, as it is only fractional on edges not mentioned by R . Theorem 5.
Let ω be an odd charging of H n . Ts ( H n , ω ) requires length Ω( n ) to refute in SP. roof. Fix some refutation R . We work in the inner ( n − × ( n − grid - in this grid every node hasdegree so we may use Lemma 11. In this grid we can find d := ( ⌊ n − ⌋ ) vertex-disjoint squares, andin each such square we can choose (Lemma 12) an edge mentioned in R . As these squares are vertexdisjoint the chosen edges form a matching and Lemma 11 tells us there is some essential cover R ofthe d -dimensional hypercube with | R | ≥ | L | . Now due to Theorem 4 we have | L | ∈ Ω( n ) and we aredone. Corollary 5. Ts ( H n , ω ) requires depth Ω(log( n )) to refute in SP . The
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