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arXiv · 1106.3527

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

Gregory Gutin·Eun Jung Kim·Arezou Soleimanfallah·Stefan Szeider·Anders Yeo
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Abstract

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[1]-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.

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Gregory Gutin, Eun Jung Kim, Arezou Soleimanfallah, Stefan Szeider, Anders Yeo. 2011-06-17. Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming. https://arxiv.org/abs/1106.3527

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