The Tractability of CSP Classes Defined by Forbidden Patterns

Abstract

The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NP-hard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions on the hypergraph of constraint scopes) and relational properties (restrictions on the language of constraint relations). Recently, some authors have considered hybrid properties that restrict the constraint hypergraph and the relations simultaneously.Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic subproblems). We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns. We show that this framework generalises relational properties and allows us to capture known hybrid tractable classes.Although we are not close to obtaining a dichotomy concerning the tractability of general forbidden patterns, we are able to make some progress in a special case: classes of problems that arise when we can only forbid binary negative patterns (generic sub-problems in which only inconsistent tuples are specified). In this case we are able to characterise very large classes of tractable and NP-hard forbidden patterns. This leaves the complexity of just one case unresolved and we conjecture that this last case is tractable.

Tractability in constraint satisfaction problems: a survey

“…It can be identified with its coloured microstructure which can be viewed as a graph over a three-valued logic in which each pair of vertices is an edge, a nonedge or unknown. A pattern P occurs in a binary instance I if there is a homomorphism from P to an induced subgraph of the coloured microstructure which preserves edges and non-edges [47]. A class of binary CSP instance can be defined by forbidding a pattern P : CSP(P ) is the set of instances in which P does not occur.…”

Beyond Consistency and Substitutability

“…Previously, we have studied different forms of forbidden patterns [2,3,4]. Forbidding a flat pattern on assignments A corresponds to a rule Q(A var ∪ A val )f (E(A)) where all quantifiers in Q are ∀ and the function f is a clause.…”

On Maximum Weight Clique Algorithms, and How They Are Evaluated

“…The reduction of constraint satisfaction problems to finding a clique in a corresponding microstructure graph is primarily studied for its theoretical properties [13][14][15][16]28]. For problems with a special objective function, a reduction to the maximum clique problem which preserves the objective value is possible-indeed, recent experimental work shows that this encoding, rather than conventional constraint programming, is the best practical approach for solving the maximum common subgraph problem when vertex or edge labels are present [39].…”