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 n 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

Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. 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. The main result of the present paper 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 n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. 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 CNF's

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

Instance compression of parametric problems and related hierarchies

We define instance compressibility ([13, 17]) for parametric problems in the classes PH and PSPACE.We observe that the problem ƩiCIRCUITSAT of deciding satisfiability of a quantified Boolean circuit with i-1 alternations of quantifiers starting with an existential quantifier is complete for parametric problems in the class Ʃp/i with respect to w-reductions, and that analogously the problem QBCSAT (Quantified Boolean Circuit Satisfiability) is complete for parametric problems in PSPACE with respect to w-reductions. We show the following results about these problems: 1. If CIRCUITSAT is non-uniformly compressible within NP, then ƩiCIRCUITSAT is non-uniformly compressible within NP, for any i≥1. 2. If QBCSAT is non-uniformly compressible (or even if satisfiability of quantified Boolean CNF formulae is non-uniformly compressible), then PSPACE ⊆ NP/poly and PH collapses to the third level. Next, we define Succinct Interactive Proof (Succinct IP) and by adapting the proof of IP = PSPACE ([11, 6]) , we show that QBCNFSAT (Quantified Boolean Formula (in CNF) Satisfiability) is in Succinct IP. On the contrary if QBCNFSAT has Succinct PCPs ([32]) , Polynomial Hierarchy (PH) collapses. After extending the notion of instance compression to higher classes, we study the hierarchical structure of the parametric problems with respect to compressibility. For that purpose, we extend the existing definition of VC-hierarchy ([13]) to parametric problems. After that, we have considered a long list of natural NP problems and tried to classify them into some level of VC-hierarchy. We have shown some of the new w-reductions in this context and pointed out a few interesting results including the ones as follows. 1. CLIQUE is VC1-complete (using the results in [14]). 2. SET SPLITTING and NAE-SAT are VC2-complete. We have also introduced a new complexity class VCE in this context and showed some hardness and completeness results for this class. We have done a comparison of VC-hierarchy with other related hierarchies in parameterized complexity domain as well. Next, we define the compression of counting problems and the analogous classification of them with respect to the notion of instance compression. We define #VC-hierarchy for this purpose and similarly classify a large number of natural counting problems with respect to this hierarchy, by showing some interesting hardness and completeness results. We have considered some of the interesting practical problems as well other than popular NP problems (e.g., #MULTICOLOURED CLIQUE, #SELECTED DOMINATING SET etc.) and studied their complexity for both decision and counting version. We have also considered a large variety of circuit satisfiability problems (e.g., #MONOTONE WEIGHTED-CNFSAT, #EXACT DNF-SAT etc.) and proved some interesting results about them with respect to the theory of instance compressibility

Manipulation of elections by minimal coalitions

Social choice is the study of the issues arising when a population of individuals attempts to combine its views with the objective of determining a collective policy. Recent research in artificial intelligence raises concerns of articial intelligence agents applying computational resources to attack an election. If we think of voting as a way to combine honest preferences, it would be undesirable for some voters cast ballots that differ from their true preferences and achieve a better result for themselves at the expense of the general social welfare. Such an attack is called manipulation. The Gibbard-Satterthwaite theorem holds that all reasonable voting rules will admit a situation in which some voter achieves a better result for itself by misrepresenting its preferences. Bartholdi and Orlin showed that finding a beneficial manipulation under the single transferable vote rule is NP-Complete. Our work explores the practical dificulty of the coalitional manipulation problem. We computed the minimum sizes of successful manipulating coalitions, and compared this to theoretical results

Parameterized aspects of team-based formalisms and logical inference

Parameterized complexity is an interesting subfield of complexity theory that has received a lot of attention in recent years. Such an analysis characterizes the complexity of (classically) intractable problems by pinpointing the computational hardness to some structural aspects of the input. In this thesis, we study the parameterized complexity of various problems from the area of team-based formalisms as well as logical inference. In the context of team-based formalism, we consider propositional dependence logic (PDL). The problems of interest are model checking (MC) and satisfiability (SAT). Peter Lohmann studied the classical complexity of these problems as a part of his Ph.D. thesis proving that both MC and SAT are NP-complete for PDL. This thesis addresses the parameterized complexity of these problems with respect to a wealth of different parameterizations. Interestingly, SAT for PDL boils down to the satisfiability of propositional logic as implied by the downwards closure of PDL-formulas. We propose an interesting satisfiability variant (mSAT) asking for a satisfiable team of size m. The problem mSAT restores the ‘team semantic’ nature of satisfiability for PDL-formulas. We propose another problem (MaxSubTeam) asking for a maximal satisfiable team if a given team does not satisfy the input formula. From the area of logical inference, we consider (logic-based) abduction and argumentation. The problem of interest in abduction (ABD) is to determine whether there is an explanation for a manifestation in a knowledge base (KB). Following Pfandler et al., we also consider two of its variants by imposing additional restrictions over the size of an explanation (ABD and ABD=). In argumentation, our focus is on the argument existence (ARG), relevance (ARG-Rel) and verification (ARG-Check) problems. The complexity of these problems have been explored already in the classical setting, and each of them is known to be complete for the second level of the polynomial hierarchy (except for ARG-Check which is DP-complete) for propositional logic. Moreover, the work by Nord and Zanuttini (resp., Creignou et al.) explores the complexity of these problems with respect to various restrictions over allowed KBs for ABD (ARG). In this thesis, we explore a two-dimensional complexity analysis for these problems. The first dimension is the restrictions over KB in Schaefer’s framework (the same direction as Nord and Zanuttini and Creignou et al.). What differentiates the work in this thesis from an existing research on these problems is that we add another dimension, the parameterization. The results obtained in this thesis are interesting for two reasons. First (from a theoretical point of view), ideas used in our reductions can help in developing further reductions and prove (in)tractability results for related problems. Second (from a practical point of view), the obtained tractability results might help an agent designing an instance of a problem come up with the one for which the problem is tractable