, Volume 64, Issue 1, pp 112–125 | Cite as

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


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 UV, 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 \(\text {\normalfont W[1]}\)-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 Global constraint Fixed-parameter tractable 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Gregory Gutin
    • 1
  • Eun Jung Kim
    • 2
  • Arezou Soleimanfallah
    • 1
  • Stefan Szeider
    • 3
  • Anders Yeo
    • 1
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonEghamUK
  2. 2.AlGCo project-team, CNRSLIRMMMontpellierFrance
  3. 3.Institute of Information SystemsVienna University of TechnologyViennaAustria

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