Tractable Optimization Problems through Hypergraph-Based Structural Restrictions

Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modelling those scenarios where solutions are associated with some given costs. Within these frameworks computing an optimal solution is an NP-hard problem in general; yet, when restricted over classes of instances whose constraint interactions can be modelled via (nearly-)acyclic graphs, this problem is known to be solvable in polynomial time. In this paper, larger classes of tractable instances are singled out, by discussing solution approaches based on exploiting hypergraph acyclicity and, more generally, structural decomposition methods, such as (hyper)tree decompositions

Proteus: A Hierarchical Portfolio of Solvers and Transformations

In recent years, portfolio approaches to solving SAT problems and CSPs have become increasingly common. There are also a number of different encodings for representing CSPs as SAT instances. In this paper, we leverage advances in both SAT and CSP solving to present a novel hierarchical portfolio-based approach to CSP solving, which we call Proteus, that does not rely purely on CSP solvers. Instead, it may decide that it is best to encode a CSP problem instance into SAT, selecting an appropriate encoding and a corresponding SAT solver. Our experimental evaluation used an instance of Proteus that involved four CSP solvers, three SAT encodings, and six SAT solvers, evaluated on the most challenging problem instances from the CSP solver competitions, involving global and intensional constraints. We show that significant performance improvements can be achieved by Proteus obtained by exploiting alternative view-points and solvers for combinatorial problem-solving.Comment: 11th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. The final publication is available at link.springer.co

Even Delta-Matroids and the Complexity of Planar Boolean CSPs

The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even $\Delta$-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Using a reduction to even $\Delta$-matroids, we then extend the tractability result to larger classes of $\Delta$-matroids that we call efficiently coverable. It properly includes classes that were known to be tractable before, namely co-independent, compact, local, linear and binary, with the following caveat: we represent $\Delta$-matroids by lists of tuples, while the last two use a representation by matrices. Since an $n\times n$ matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary $\Delta$-matroids.Comment: 33 pages, 9 figure

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$