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
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6478)


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.


General Factor Bipartite Graph Parameterized Complexity Global Constraint Reduction Rule 
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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gregory Gutin
    • 1
  • Eun Jung Kim
    • 1
  • Arezou Soleimanfallah
    • 1
  • Stefan Szeider
    • 2
  • Anders Yeo
    • 1
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonEgham, SurreyUK
  2. 2.Institute of Information SystemsVienna University of TechnologyViennaAustria

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