Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming

摘要

The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U (+) V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint.We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by | V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes W[l]-hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic in­stances, each of which can be solved in polynomial time by dynamic programming.