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

TL;DR

This paper investigates the parameterized complexity of the general factor problem in bipartite graphs, showing fixed-parameter tractability under certain list constraints and establishing hardness results otherwise, with applications to constraint programming.

Contribution

It proves fixed-parameter tractability for bipartite general factors when vertex lists are of length 1, and W[1]-hardness when lists are of length 2, advancing understanding in parameterized complexity.

Findings

FPT when all vertices in U have lists of size 1

W[1]-hardness when vertices in U have lists of size 2

Reduction to acyclic instances enables polynomial-time solutions

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.