Massimo Lauria

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.

Parameterized Bounded-Depth Frege Is not Optimal

A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution---a parameterized version of classical Resolution---and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size nΩ(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.

Optimality of size-degree tradeoffs for polynomial calculus

There are methods to turn short refutations in <i>polynomial calculus</i> (Pc) and <i>polynomial calculus with resolution</i> (Pcr) into refutations of low degree. Bonet and Galesi [1999, 2003] asked if such size-degree tradeoffs for Pc [Clegg et al. 1996; Impagliazzo et al. 1999] and Pcr [Alekhnovich et al. 2004] are optimal. We answer this question by showing a polynomial encoding of the <i>graph ordering principle</i> on <i>m</i> variables which requires Pc and Pcr refutations of degree Ω(&sqrt; <i>m</i>). Tradeoff optimality follows from our result and from the short refutations of the graph ordering principle in Bonet and Galesi [1999, 2001]. We then introduce the algebraic proof system Pcr<i>_k</i> which combines together polynomial calculus and <i>k-DNF resolution</i> (Res<i>_k</i>). We show a size hierarchy theorem for Pcr<i>_k</i>: Pcr<i>_k</i> is exponentially separated from Pcr<i>_k+1</i>. This follows from the previous degree lower bound and from techniques developed for Res<i>_k</i>. Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in Pcr<i>_k</i>.

A characterization of tree-like Resolution size

We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas [BGL11] and for the classical pigeonhole principle [BGL10]. The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i. e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

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 which 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 nO(k) steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas.

A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover-Delayer games

In this note we show that the asymmetric Prover-Delayer game developed in Beyersdorff et al. (2010) [2] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form 2^@W^(^n^l^o^g^n^) for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki (1999) [7] and Dantchev and Riis (2001) [5].

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a “black-box” technique for proving space lower bounds via a “static” complexity measure that works against any resolution refutation—previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

Minimum-Energy Broadcast and disk cover in grid wireless networks

The Minimum Energy Broadcast problem consists in finding the minimum-energy range assignment for a given set S of n stations of an ad hoc wireless network that allows a source station to perform broadcast operations over S We prove a nearly tight asymptotical bound on the optimal cost for the Minimum Energy Broadcast problem on square grids. We emphasize that finding tight bounds for this problem restriction is far to be easy: it involves the Gausss Circle problem and the Apollonian Circle Packing. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about π and it yields

Divide and conquer is almost optimal for the bounded-hop MST problem on random euclidean instances

\Omega(\sqrt{n})

On the Automatizability of Polynomial Calculus

hops As a by product, we get nearly tight bounds for the Minimum Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius Finally, we emphasize that our upper bounds are obtained via polynomial time constructions