Parameterized Complexity of the MINCCA Problem on Graphs of Bounded Decomposability

Abstract

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence ($\textsc {MinCCA}$) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [14] that the $\textsc {MinCCA}$ problem is $\mathsf{FPT}$ when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for $\textsc {MinCCA}$:

• the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of [14];

• it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph;

• it is $\mathsf{FPT}$ parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the $\mathsf{FPT}$ result given in [14];

• it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.

This work is supported by the bilateral research program of CNRS and TUBITAK under grant no.114E731.

M. Shalom—The work of this author is supported in part by the TUBITAK 2221 Programme.