Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the way up to $n^{\delta}$ for some universal constant$\delta$. This shows that the $n^{O(d)}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining $\mathsf{NP}$-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively

Narrow Proofs May Be Maximally Long

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^Omega(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. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w

Automating Resolution is NP-hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show that it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless~NP is included in SUBEXP or QP, respectively.Peer ReviewedPostprint (author's final draft

New Results on Cutting Plane Proofs for Horn Constraint Systems

In this paper, we investigate properties of cutting plane based refutations for a class of integer programs called Horn constraint systems (HCS). Briefly, a system of linear inequalities A * x >= b is called a Horn constraint system, if each entry in A belongs to the set {0,1,-1} and furthermore there is at most one positive entry per row. Our focus is on deriving refutations i.e., proofs of unsatisfiability of such programs using cutting planes as a proof system. We also look at several properties of these refutations. Horn constraint systems can be considered as a more general form of propositional Horn formulas, i.e., CNF formulas with at most one positive literal per clause. Cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of a pair of inference rules. These are called the addition rule (ADD) and the division rule (DIV). In this paper, we show that cutting plane calculus is still complete for Horn constraints when every intermediate constraint is required to be Horn. We also investigate the lengths of cutting plane proofs for Horn constraint systems

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing

Analyzing Satisfiability and Refutability in Selected Constraint Systems

This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both Boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining Boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability