Soft constraints are functions returning costs, and are essential in modeling over-constrained and optimization problems. We are interested in tackling soft constrained problems with adversarial conditions. Aiming at generalizing the weighted and quantified constraint satisfaction frameworks, a Quantified Weighted Constraint Satisfaction Problem (QWCSP) consists of a set of finite domain variables, a set of soft constraints, and a min or max quantifier associated with each of these variables. We formally define QWCSP, and propose a complete solver which is based on alpha-beta pruning. QWCSPs are useful special cases of QCOP/QCOP+, and can be solved as a QCOP/QCOP+. Restricting our attention to only QWCSPs, we show empirically that our proposed solving techniques can better exploit problem characteristics than those developed for QCOP/QCOP+. Experimental results confirm the feasibility and efficiency of our proposals.

This paper considers matrix models, a class of CSPs which generally exhibit significant symmetries. It proposed the idea of LexLeader feasibility checkers that verify, during search, whether the current partial assignment can be extended into a canonical solution. The feasibility checkers are based on a novel result by [Katsirelos et al., 2010] on how to check efficiently whether a solution is canonical. The paper generalizes this result to partial assignments, various variable orderings, and value symmetries. Empirical results on 5 standard benchmarks shows that feasibility checkers may bring significant performance gains, when jointly used with DOUBLELEX or SNAKELEX.