Parameterized Complexity of DPLL Search Procedure

Zusammenfassung: 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 n Ω(k) steps for a non-trivial distribution of graphs close to the critical threshold. These results are presented in the general framework for parameterized proof complexity as introduced by Dantchev, Martin, and Szeider [3]. There the authors concentrate on tree-like Parameterized Resolution—a parameterized version of classical Resolution—and their gap complexity theorem implies lower bounds for that system. In our paper [2] we significantly improve 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 n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. Zusammenfassung: We study popular local search and greedy algorithms for scheduling with unrestricted related machines. The performance guarantees of these algorithms are well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable if they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. We show that the smoothed performance guarantee both of the lex-jump algorithm and of the list scheduling algorithm is Θ(log ϕ) where 1/ϕ is a parameter measuring the magnitude of the perturbation. This bound contrasts with the worst-case bounds that depend on the number of machines. The bound for the lex-jump algorithm immediately implies that also the smoothed price of anarchy for routing games on parallel links is Θ(log ϕ).