Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. RANK COMPLEXITY GAP FOR LOVÁSZ-SCHRIJVER AND SHERALI-ADAMS PROOF SYSTEMS Stefan Dantchev and Barnaby MartinAbstract. We prove a dichotomy theorem for the rank of propositional contradictions, uniformly generated from first-order sentences, in both the Lovász-Schrijver (LS) and Sherali-Adams (SA) refutation systems. More precisely, we first show that the propositional translations of firstorder formulae that are universally false, i.e. fail in all finite and infinite models, have LS proofs whose rank is constant, independent of the size of the (finite) universe. In contrast to that, we prove that the propositional formulae that fail in all finite models, but hold in some infinite structure, require proofs whose SA rank grows polynomially with the size of the universe. Until now, this kind of so-called "complexity gap" theorem has been known for tree-like Resolution and, in somehow restricted forms, for the Resolution and Nullstellensatz systems. As far as we are aware, this is the first time the Sherali-Adams lift-and-project method has been considered as a propositional refutation system (since the conference version of this paper, SA has been considered as a refutation system in several further papers). An interesting feature of the SA system is that it simulates LS, the Lovász-Schrijver refutation system without semidefinite cuts, in a rank-preserving fashion.

On the Rank of Cutting-Plane Proof Systems

“…For example, it is known that the Gomory-Chvátal rank of a polytope contained in the n-dimensional 0/1-cube is at most O(n 2 log n) Eisenbrand and Schulz [2003], whereas the rank of all other methods mentioned before is bounded above by n, which is known to be tight (see, e.g., Cook and Dash [2001], Cornuéjols [2008]). These convexification procedures can also be viewed as propositional proof systems (e.g., Chvátal et al [1989], Dantchev [2007], Dash [2005]), each using its own set of rules to prove that a system of linear inequalities with integer coefficients cover, and sparsest cut, and in Mathieu and Sinclair [2009] integrality gaps for the fractional matching polytope, which has Gomory-Chvátal rank 1, are provided, showing that although the matching problem can be solved in polynomial time, it cannot be approximated well with a small number of rounds of the Sherali-Adams operator. In addition, it was shown that for certain tautologies that can be expressed in first-order logic, the Lovász-Schrijver N + rank can be constant, whereas the Sherali-Adams rank grows poly-logarithmically Dantchev [2007].…”

“…These convexification procedures can also be viewed as propositional proof systems (e.g., Chvátal et al [1989], Dantchev [2007], Dash [2005]), each using its own set of rules to prove that a system of linear inequalities with integer coefficients cover, and sparsest cut, and in Mathieu and Sinclair [2009] integrality gaps for the fractional matching polytope, which has Gomory-Chvátal rank 1, are provided, showing that although the matching problem can be solved in polynomial time, it cannot be approximated well with a small number of rounds of the Sherali-Adams operator. In addition, it was shown that for certain tautologies that can be expressed in first-order logic, the Lovász-Schrijver N + rank can be constant, whereas the Sherali-Adams rank grows poly-logarithmically Dantchev [2007]. A link between the Sherali-Adams closure and border bases, and hence algebraic geometry, has been established in Pokutta and Schulz [2009].…”

The Limits of Tractability in Resolution-Based Propositional Proof Systems

“…Moreover, the hard Case 2 prevails exactly when φ has some (infinite) model. Since the publication of [1], another gap has been discovered based on rank, and not size, for the integer linear programming-based refutation systems of Lovász-Schrijver and Sherali-Adams [2]. In these cases, the separating criterion is again whether or not an infinite model exists for φ, with the hard case -of polynomial instead of constant rank -prevailing if it does.…”

Cutting Planes and the Parameter Cutwidth

“…A series of papers in recent years has studied so-called gap phenomena in propositional refutation systems. This began with [12], and has continued with, e.g., [7,13]. A complexity gap is given for two other refutation systems based on ILP-namely those of Lovász-Schrijver and Sherali-Adams (SA)-in [7].…”

“…This began with [12], and has continued with, e.g., [7,13]. A complexity gap is given for two other refutation systems based on ILP-namely those of Lovász-Schrijver and Sherali-Adams (SA)-in [7]. In each case the relevant parameter is rank and, as in [12], the separating criterion is whether or not ψ has some (infinite) model.…”