Nicola Galesi

Space complexity of random formulae in resolution

We study the space complexity of refuting unsatisfiable random k-CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any e > 0, with high probability a random k-CNF over n variables and Δn clauses requires resolution clause space of Ω(n/Δ1 + e). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density Δ > √n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with Δn clauses requires treelike refutation size of exp(Ω(n/Δ1 + e). for any e > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipanite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G.

On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only superpolynomial separations were known [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467; J. Johannsen, Inform. Process. Lett., 67 (1998), pp. 37--41; P. Clote and A. Setzer, in Proof Complexity and Feasible Arithmetics, Amer. Math. Soc., Providence, RI, 1998, pp. 93--117]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [ Combinatorica, 19 (1999), pp. 403--435] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually, this last separation also provides a separation between tree-like resolution and ordered resolution, and thus the corresponding superpolynomial separation of [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [A. Goerdt, Ann. Math. Artificial Intelligence, 6 (1992), pp. 169--184].

Rank bounds and integrality gaps for cutting planes procedures

We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovasz and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: first, we prove near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank resolution proofs. Thirdly, we give examples where the CP rank is O(log n) but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomial-size CP/LS proofs, but requiring linear rank.

A study of proof search algorithms for resolution and polynomial calculus

The paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof systems: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of E. Ben-Sasson and A. Wigderson (1999) for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship, referred to as size-width trade-off. We moreover obtain the optimality of the size width trade-off for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive and linear. As for the second system, we show that the direct translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than /spl Omega/ (log n). A consequence of this is that the simulation of resolution by PC of M. Clegg, J. Edmonds and R. Impagliazzo (1996) cannot be improved to better than quasipolynomial in the case where we start with small resolution proofs. We conjecture that the simulation of M. Clegg et al. is optimal.

On the complexity of resolution with bounded conjunctions

We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Krajicek in (Fund. Math. 170 (1-3) (2001) 123) which extend Resolution by allowing disjunctions of conjunctions of up to k ≥ 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover Resolution, while simulating treelike Res(k), is almost exponentially separated from treelike Res(k). To study space complexity for general Res(k) we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for Resolution and to extend them to Res(k).

Optimality of size-width tradeoffs for resolution

Abstract.This paper is concerned with the complexity of proofs and of searching for proofs in resolution. We show that the recently proposed algorithm of Ben-Sasson & Wigderson for searching for proofs in resolution cannot give better than weakly exponential performance. This is a consequence of our main result: we show the optimality of the general relationship called size-width tradeoffs in Ben-Sasson & Wigderson. Moreover we obtain the optimality of the size-width tradeoffs for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive.

Proofs of Space: When Space Is of the Essence

Proofs of computational effort were devised to control denial of service attacks. Dwork and Naor (CRYPTO ’92), for example, proposed to use such proofs to discourage spam. The idea is to couple each email message with a proof of work that demonstrates the sender performed some computational task. A proof of work can be either CPU-bound or memory-bound. In a CPU-bound proof, the prover must compute a CPU-intensive function that is easy to check by the verifier. A memory-bound proof, instead, forces the prover to access the main memory several times, effectively replacing CPU cycles with memory accesses.

Exponential separations between restricted resolution and cutting planes proof systems

We prove an exponential lower bound for tree-like cutting planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both cutting planes and resolution; in both cases only superpolynomial separations were known before. In order to prove this, we extend the lower bounds on the depth of monotone circuits of R. Raz and P. McKenzie (1997) to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution front (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Puatam resolution proof. Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separations was previously known.

Parameterized Complexity of DPLL Search Procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game that models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires nΩ(k) steps for a nontrivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked by Beyersdorff et al. [2012] of understanding the Resolution complexity of this family of formulas.

Monotone simulations of nonmonotone proofs

We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis /spl and/, V) can be turned into a proof containing only monotone formulas of size m/sup O(log m)/ and with the number of proof lines polynomial in m. Also we show that some interesting special cases, namely the functional and the onto versions of PHP and a version of the matching principle, have polynomial size monotone proofs.