Parameterized complexity of complete multipartite hosts

Determine whether Feasible Matching-Match on complete k-partite host graphs is fixed-parameter tractable or W[1]-hard when parameterized by the number k of parts.

Background

The paper establishes that Feasible Matching-Match (FMMP) on complete k-partite graphs is solvable in XP when parameterized by the number k of parts. For every fixed k, the authors provide a polynomial-time algorithm that permits arbitrary precoloring and an unrestricted number of colors. However, the dependence on k is not shown to be fixed-parameter tractable, and the paper does not establish matching parameterized hardness. The unresolved problem is therefore to determine the precise parameterized complexity of FMMP for complete k-partite host graphs with k as the parameter.