Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint ProgrammingGregory Gutin, Eun Jung Kim, Arezou Soleimanfallah, , and Anders Yeo.Algorithmica vol. 64, no. 1, pp. 112-125, 2012. Prelimininary and shortened version appeared in the proceedings of IPEC 2010, International Symposium on Parameterized and Exact Computation (formerly IWPEC), December 13-15, 2010, IMSc, Chennai, India. Lecture Notes in Computer Science, vol. 6478, pp. 158-169, 2010. 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. Peprint available from [arXiv:1106.3527] |