Exponential lower bounds and Integrality Gaps for Tree-like Lovász-Schrijver Procedures

Abstract

The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zero-one programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems, and to solve certain instances of Boolean satisfiability.Our first result is a size/rank tradeoff for tree-like Lovász-Schrijver refutations, showing that any refutation that has small size also has small rank. This allows us to immediately derive exponential size lower bounds for tree-like refutations of many unsatisfiable systems of inequalities where prior to our work, only strong rank bounds were known.Unfortunately, we show that this tradeoff does not hold more generally for derivations of arbitrary inequalities. We give a very simple example showing that derivations can be very small but nonetheless require maximal rank. This rules out a generic argument for obtaining a size-based integrality gap from the corresponding rankbased integrality gap. Our second contribution is to show that a modified argument can often be used to prove size-based integrality gaps from rank-based integrality gaps. We apply this method to prove size-based integrality gaps for several prominant examples where prior to our work, only rank-based integrality gaps were known.Our third contribution is to prove new separation results. Using our machinery for converting rank-based lower bounds and integrality gaps into size-based lower bounds, we show that tree-like LS + cannot polynomially simulate tree-like Cutting Planes, and that tree-like LS + cannot polynomially simulate resolution.We conclude by examining size/rank tradeoffs be- * Research supported by NSERC.yond the LS systems. We show that for Shirali-Adams and Lasserre systems, size/rank tradeoffs continue to hold, even in the general (non-tree) case. A full version of this paper is available at the Electronic Colloquium on Computational Complexity [23].

Circular (Yet Sound) Proofs

“…As a consequence to this, exponential length lower bounds follow from linear width lower bounds for Circular Resolution, or equivalently, from linear degree lower bounds for Sherali-Adams. In particular, since linear degree lower bounds for Sherali-Adams are known for 3-CNF formulas with expanding incidence graphs (see [14,20]), we get the following: Corollary 15. There are families of 3-CNF formulas (F n ) n≥1 , where F n has O(n) variables and O(n) clauses, such that every Circular Resolution refutation of F n has width Ω(n) and size 2 Ω(n) .…”

“…Yet another consequence of the equivalence between Circular Resolution and Sherali-Adams is that Circular Resolution has a length-width relationship in the style of the one due to Ben-Sasson and Wigderson for Dag-like Resolution [5]. This follows from Theorem 7 in combination with the size-degree relationship that is known to hold for Sherali-Adams (see [20,2]). As a consequence to this, exponential length lower bounds follow from linear width lower bounds for Circular Resolution, or equivalently, from linear degree lower bounds for Sherali-Adams.…”

“…In its original presentation, the Sherali-Adams hierarchy can already be thought of as a proof system for reasoning with polynomial inequalities, with the levels of the hierarchy corresponding to the degrees of the polynomials. For propositional logic, the system was studied in [10], and developed further in [20,3]. Those works consider the version of the proof system in which each propositional variable X comes with a formal twin variableX, that is to be interpreted by the negation of X.…”

Sherali-adams relaxations of the matching polytope

“…A recent paper of Pitassi and Segerlind [23] proves exponential size lower bounds for tree-like LS + proofs of unsatisfiability for several important classes of CNFs, and also shows that tree-like LS + proofs cannot efficiently simulate certain other standard proof systems. This differs from the aforementioned work in that the lower bounds are for the size of the proofs rather than for the rank (which corresponds to depth in the tree-like scenario).…”

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

“…For tree-like LS and LS + what happens is closer to what happens for dag-like resolution. The analogue size-rank tradeoff for tree-like LS and LS + was shown by Pitassi and Segerlind [36]: if a system of linear inequalities with n variables has a tree-like LS-refutation of size s, then it also has an LS k -refutation with k = O( √ n log s), and the same for LS + . Again this gives an algorithm that, given a system of linear inequalities that is contradictory over {0, 1} n , finds an LS-refutation in time n O( √ n log s) , where s is the size of the shortest tree-like LS-refutation, and n is the number of variables.…”