Tractable cases of the extended global cardinality constraint

We study the consistency and domain consistency problem for extended global cardinality (EGC) constraints. An EGC constraint consists of a set  X of variables, a set  D of values, a domain  for each variable  x , and a “cardinality set”  K ( d ) of non-negative integers for each value  d . The probl...

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Vydáno v:Constraints : an international journal Ročník 16; číslo 1; s. 1 - 24
Hlavní autoři: Samer, Marko, Szeider, Stefan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.01.2011
Springer
Témata:
Applied sciences
Artificial Intelligence
Computer Science
Computer science; control theory; systems
Exact sciences and technology
Flows in networks. Combinatorial problems
Information retrieval. Graph
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimization
Theoretical computing
Domain consistency
Bounded treewidth
Global constraints
General factor problem
Parameterized complexity
Constraint satisfaction
Linear time
Treewidth
Constrained optimization
Polynomial time
Multiplicity
NP complete problem
Bipartite graph
Graph matching
ISSN:1383-7133, 1572-9354
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Popis
Shrnutí:We study the consistency and domain consistency problem for extended global cardinality (EGC) constraints. An EGC constraint consists of a set  X of variables, a set  D of values, a domain  for each variable  x , and a “cardinality set”  K ( d ) of non-negative integers for each value  d . The problem is to instantiate each variable  x with a value in  D ( x ) such that for each value  d , the number of variables instantiated with  d belongs to the cardinality set  K ( d ). It is known that this problem is NP-complete in general, but solvable in polynomial time if all cardinality sets are intervals. First we pinpoint connections between EGC constraints and general factors in graphs. This allows us to extend the known polynomial-time case to certain non-interval cardinality sets. Second we consider EGC constraints under restrictions in terms of the treewidth of the value graph (the bipartite graph representing variable-value pairs) and the cardinality-width (the largest integer occurring in the cardinality sets). We show that EGC constraints can be solved in polynomial time for instances of bounded treewidth, where the order of the polynomial depends on the treewidth. We show that (subject to the complexity theoretic assumption  FPT ≠ W[1]) this dependency cannot be avoided without imposing additional restrictions. If, however, also the cardinality-width is bounded, this dependency gets removed and EGC constraints can be solved in linear time.
ISSN:1383-7133
1572-9354
DOI:10.1007/s10601-009-9079-y