The Complexity of Valued Constraint Satisfaction Problems in a Nutshell

National audienceThe valued constraint satisfaction problem was introduced by Schiex et al. [23] as a unifying framework for studying constraint programming with soft constraints. A systematic worst-case complexity theoretical investigation of this problem was initiated by Cohen et al. [4], building on ideas from the successful classi cation programme for the ordinary constraint satisfaction problem. In addition to the decision problem for constraint satisfaction, this framework also captures problems as varied as Max CSP and integer programming with bounded domains. This paper is intended to give a quick introduction to the questions, the main results, and the current state of the complexity classi cation of valued constraint satisfaction problems. Two special cases are looked at in some detail : the classi cation for the Boolean domain and the less well-understood case of Max CSP. Some recent results for general constraint languages are also reviewed, as well as the connection to the very active study of approximation algorithms for Max CSP

Constraint Satisfaction Techniques for Combinatorial Problems

The last two decades have seen extraordinary advances in tools and techniques for constraint satisfaction. These advances have in turn created great interest in their industrial applications. As a result, tools and techniques are often tailored to meet the needs of industrial applications out of the box. We claim that in the case of abstract combinatorial problems in discrete mathematics, the standard tools and techniques require special considerations in order to be applied effectively. The main objective of this thesis is to help researchers in discrete mathematics weave through the landscape of constraint satisfaction techniques in order to pick the right tool for the job. We consider constraint satisfaction paradigms like satisfiability of Boolean formulas and answer set programming, and techniques like symmetry breaking. Our contributions range from theoretical results to practical issues regarding tool applications to combinatorial problems. We prove search-versus-decision complexity results for problems about backbones and backdoors of Boolean formulas. We consider applications of constraint satisfaction techniques to problems in graph arrowing (specifically in Ramsey and Folkman theory) and computational social choice. Our contributions show how applying constraint satisfaction techniques to abstract combinatorial problems poses additional challenges. We show how these challenges can be addressed. Additionally, we consider the issue of trusting the results of applying constraint satisfaction techniques to combinatorial problems by relying on verified computations

The Complexity of Rooted Phylogeny Problems

Several computational problems in phylogenetic reconstruction can be formulated as restrictions of the following general problem: given a formula in conjunctive normal form where the literals are rooted triples, is there a rooted binary tree that satisfies the formula? If the formulas do not contain disjunctions, the problem becomes the famous rooted triple consistency problem, which can be solved in polynomial time by an algorithm of Aho, Sagiv, Szymanski, and Ullman. If the clauses in the formulas are restricted to disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem remains NP-complete. We systematically study the computational complexity of the problem for all such restrictions of the clauses in the input formula. For certain restricted disjunctions of triples we present an algorithm that has sub-quadratic running time and is asymptotically as fast as the fastest known algorithm for the rooted triple consistency problem. We also show that any restriction of the general rooted phylogeny problem that does not fall into our tractable class is NP-complete, using known results about the complexity of Boolean constraint satisfaction problems. Finally, we present a pebble game argument that shows that the rooted triple consistency problem (and also all generalizations studied in this paper) cannot be solved by Datalog

The complexity of conservative finite-valued CSPs

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. We consider the case of so-called \emph{conservative} languages; that is, languages containing all unary cost functions, thus allowing arbitrary restrictions on the domains of the variables. This problem has been studied by Bulatov [LICS'03] for $\{0,\infty\}$-valued languages (i.e. CSP), by Cohen~\etal\ (AIJ'06) for Boolean domains, by Deineko et al. (JACM'08) for $\{0,1\}$-valued cost functions (i.e. Max-CSP), and by Takhanov (STACS'10) for $\{0,\infty\}$-valued languages containing all finite-valued unary cost functions (i.e. Min-Cost-Hom). We give an elementary proof of a complete complexity classification of conservative finite-valued languages: we show that every conservative finite-valued language is either tractable or NP-hard. This is the \emph{first} dichotomy result for finite-valued VCSPs over non-Boolean domains.Comment: 15 page