Solving Over-Constrained Problems with SAT

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We have defined a new formalism, based on Max-SAT, for encoding and solving over-constrained problems. Our formalism is an extension of Boolean CNF formulas in which we deal with blocks of Boolean clauses instead of dealing with individual clauses. Every block, formed by a set of clauses, is declared either as a hard block (i.e., must be satisfied by any solution) or as a soft block (i.e., can be violated by some solution). The idea behind the notion of block is that it encodes a problem constraint (for example, adjacent vertices have different colors); in general, it is not enough a single clause to encode a problem constraint. We call soft CNF formulas to this new kind of formulas.