Parameterized Complexity of the MINCCA Problem on Graphs of Bounded Decomposability

Abstract

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 (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 MinCCA problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for 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 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 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.

Parameterized complexity of finding a spanning tree with minimum reload cost diameter

“…Note that in the above reduction the cost function c does not satisfy the triangle inequality at vertices p i or n i for i ∈ [ n ], and recall that this is unavoidable since otherwise the problem would be polynomial . It is worth mentioning that using the ideas in the proof of (Theorem of the full version) it can be proved that the Diameter‐Tree problem is also NP‐hard on planar graphs with Δ ≤ 4, k = 0, and a bounded number of colors; we omit the details here.…”

“…Recent works in the literature focused on numerous problems related to the reload cost concept: the minimum reload cost cycle cover problem [17], the problems of finding a path, trail or walk with minimum total reload cost between two given vertices [20], the problem of finding a spanning tree that minimizes the sum of reload costs of all paths between all pairs of vertices [18], various path, tour, and flow problems related to reload costs [2], the minimum changeover cost arborescence problem [16,22,23,25], and problems related to finding a proper edge coloring of the graph so that the total reload cost is minimized [24].…”

Minimum Reload Cost Graph Factors

“…The reload cost concept was first introduced by Wirth and Steffan [27] who focused on the problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible. Other works also focused on numerous optimization problems regarding reload costs: the problems of finding a path, trail or walk with minimum total reload cost between two given vertices [13], numerous path, tour, and flow problems [1], the minimum changeover cost arborescence problem [10,14,16,18], problems about finding a proper edge coloring of a graph such that the total reload cost is minimized [17], and the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized [12].…”

Complexity of edge coloring with minimum reload/changeover costs

“…Even within the same technology, switching between different providers, for instance switching between different commercial satellite providers in satellite networks, leads to a switching cost. All applications hitherto mentioned can be modeled using traversal costs where an edge‐colored graph is given as input, and this is the focus of the works in the literature, for example, .…”

“…Various problems about the reload cost and changeover cost concept have been studied in the literature: the minimum reload cost diameter spanning tree problem , the minimum reload cost cycle cover problem , the problem of finding a path, trail, or walk of minimum reload cost between two given vertices , the problem of finding a spanning tree that minimizes the sum of reload costs over the paths between all pairs of vertices , the problem of finding a spanning tree that minimizes the reload cost from a given root vertex to all other vertices, and finally the minimum changeover cost arborescence problem, which is to find a spanning tree that minimizes the total changeover cost from a given root vertex to all other vertices .…”