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

Gregory Gutin, Eun Jung Kim, Arezou Soleimanfallah, Stefan Szeider, Anders Yeo
2010 Lecture Notes in Computer Science  
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 NPhard 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
more » ... lem 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 fixedparameter 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.
doi:10.1007/978-3-642-17493-3_16 fatcat:6pop7th6hrebbkwgphlzhn2pau