Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordström
2016 ACM Transactions on Computational Logic  
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n O(w) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound
more » ... not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w. * This is the full-length version of the paper [ALN14], which appeared in Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC '14). 1 Introduction have a hard time competing with CDCL solvers. Proof size and space in PCR are defined in analogy with resolution, but counting monomials rather than clauses, and the measure corresponding to width of clauses is (total) degree of polynomials. It is straightforward to show that PCR can simulate resolution efficiently with respect to all of these measures, meaning that the same worst case upper bounds as in resolution apply to PCR. It was proven in [IPS99] that strong degree lower bounds imply strong size lower bounds, which is an exact analogue of the size-width relation for resolution in [BW01] discussed above, and this size-degree relation has been employed to prove exponential lower bounds on size in a number of papers, with the most general setting perhaps provided in [AR03, MN15]. The first lower bounds on space in PCR were obtained in [ABRW02], but only worked for CNF formulas of unbounded width. In [FLN + 12] the techniques in [ABRW02] were adapted to prove space lower bounds also for formulas of constant width, and optimal (linear) lower bounds on space were finally obtained in [BG13] . It is worth noting, however, that these bounds are not derived from degree lower bounds-it remains unknown whether an analogue of [AD08] holds for PCR (although [FLM + 13] has reported some progress on this and related open questions). Strong trade-offs between size and space as well as between degree and space have been shown in [BNT13], but-again in analogy with resolution-the exact relations between size and degree remains unclear. The same blow-up as in [BW01] occurs in [IPS99] when small size is converted to small degree, but it is not known whether this is necessary or just an artifact of the proof. Also, it was shown in [CEI96] that a degree upper bound of d implies proof size at most n O(d) , but it has been open whether this is tight or not. Yet another way to achieve greater expressivity than in resolution is to translate clauses into linear inequalities and manipulate them using 0-1 linear programming. Perhaps the simplest and most well-known example of this approach is the cutting planes proof system introduced in [CCT87] based on ideas in [Chv73, Gom63] . In this paper, however, we will be interested in somewhat related but different semialgebraic methods operating on linear programming relaxations of the CNF translations, such as the Sherali-Adams, Lovász-Schrijver , and Lasserre hierarchies used for attacking NP-hard optimization problems. The Sherali-Adams (SA) method [SA90] provides a hierarchy of linear programming relaxations of any given 0-1 integer program. The nth level of the hierarchy, where n is the number of 0-1 integer variables, wipes out the integrality gap and is thus exact, but also leads to an exponential blow-up in problem size. The main point of the method, however, is that any linear function of the variables can be optimized over the kth level of the hierarchy in time n O(k) , and in particular feasibility of the kth level relaxation can be checked in that time. In the context of proof complexity, what this means is that if the kth level relaxation of the integer programming formulation of a CNF formula in infeasible (the minimal such k is known as the SA rank of the integer program), then there is an n O(k) -time algorithm that can detect this. Furthermore, since the kth level of the hierarchy is an explicitly defined linear program, its infeasibility can be certified as a positive linear combination of its defining inequalities. Such a certificate is a rank-k Sherali-Adams refutation of the corresponding CNF formula. The Lovász-Schrijver approach [LS91] can be thought of as (and indeed it is formally equivalent to) an iterated version of the level-2 SA relaxation. The point is again that any linear function can be optimized over the linear program after k iterations in time n O(k) . Lovász and Schrijver also introduced a method LS + , which uses semidefinite programming instead of linear programming, and which is significantly stronger in some notable cases of interest in combinatorial optimization. The Lasserre method [Las01], finally, is basically the Sherali-Adams method with semidefinite programming conditions at all levels of the hierarchy. Again it stratifies into levels and the kth level can be solved in time n O(k) . Moreover, Lasserre's method is the strongest of all three in the sense that, level by level, it provides the tightest of all three approximations of the integer linear program. We refer to [Lau01, CT12] for a more detailed discussion of Sherali-Adams,
doi:10.1145/2898435 fatcat:fyixt7jbinastnhcaltahbrjmi