Víctor Dalmau

A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the size-width relationship is tight.

A new tractable class of constraint satisfaction problems

Abstract In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras.

The complexity of counting homomorphisms seen from the other side

For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM(C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.

Linear Datalog and Bounded Path Duality of Relational Structures

In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem

Beyond hypertree width: decomposition methods without decompositions

\csp(\best)

On the power of k -consistency

with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL.

Majority constraints have bounded pathwidth duality

The general intractability of the constraint satisfaction problem has motivated the study of restrictions on this problem that permit polynomial-time solvability. One major line of work has focused on structural restrictions, which arise from restricting the interaction among constraint scopes. In this paper, we engage in a mathematical investigation of generalized hypertree width, a structural measure that has up to recently eluded study. We obtain a number of computational results, including a simple proof of the tractability of CSP instances having bounded generalized hypertree width.

From pebble games to tractability: an ambidextrous consistency algorithm for quantified constraint satisfaction

The k-consistency algorithm for constraint-satisfaction problems proceeds, roughly, by finding all partial solutions on at most k variables and iteratively deleting those that cannot be extended to a partial solution by one more variable. It is known that if the core of the structure encoding the scopes of the constraints has treewidth at most k, then the k-consistency algorithm is always correct. We prove the exact converse to this: if the core of the structure encoding the scopes of the constraints does not have treewidth at most k, then the k-consistency algorithm is not always correct. This characterizes the exact power of the k-consistency algorithm in structural terms.

Generalized Satisfiability with Limited Occurrences per Variable: A Study through Delta-Matroid Parity

We study certain constraint satisfaction problems which are the problems of deciding whether there exists a homomorphism from a given relational structure to a fixed structure with a majority polymorphism. We show that such a problem is equivalent to deciding whether the given structure admits a homomorphism from an obstruction belonging to a certain class of structures of bounded pathwidth. This implies that the constraint satisfaction problem for any fixed structure with a majority polymorphism is in NL.

Constraint Satisfaction Problems in Non-deterministic Logarithmic Space

The constraint satisfaction problem (CSP) and quantified constraint satisfaction problem (QCSP) are common frameworks for the modelling of computational problems. Although they are intractable in general, a rich line of research has identified restricted cases of these problems that are tractable in polynomial time. Remarkably, many tractable cases of the CSP that have been identified are solvable by a single algorithm, which we call here the consistency algorithm. In this paper, we give a natural extension of the consistency algorithm to the QCSP setting, by making use of connections between the consistency algorithm and certain two-person pebble games. Surprisingly, we demonstrate a variety of tractability results using the algorithm, revealing unified structure among apparently different cases of the QCSP.