6 research outputs found
Parameterized complexity of the MINCCA problem on graphs of bounded decomposability
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 G\"oz\"upek et al. [TCS 2016] that the 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 the MINCCA problem:
- 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 G\"oz\"upek et al. [TCS 2016];
- 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 G\"oz\"upek et al. [TCS 2016];
- 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.Comment: 25 pages, 11 figure
Slim Tree-Cut Width
Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties.
In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as an approximation algorithm for computing the parameter
A new width parameter of graphs based on edge cuts: -edge-crossing width
We introduce graph width parameters, called -edge-crossing width and
edge-crossing width. These are defined in terms of the number of edges crossing
a bag of a tree-cut decomposition. They are motivated by edge-cut width,
recently introduced by Brand et al. (WG 2022). We show that edge-crossing width
is equivalent to the known parameter tree-partition-width. On the other hand,
-edge-crossing width is a new parameter; tree-cut width and
-edge-crossing width are incomparable, and they both lie between
tree-partition-width and edge-cut width.
We provide an algorithm that, for a given -vertex graph and integers
and , in time either outputs
a tree-cut decomposition certifying that the -edge-crossing width of
is at most or confirms that the -edge-crossing width
of is more than . As applications, for every fixed , we obtain
FPT algorithms for the List Coloring and Precoloring Extension problems
parameterized by -edge-crossing width. They were known to be W[1]-hard
parameterized by tree-partition-width, and FPT parameterized by edge-cut width,
and we close the complexity gap between these two parameters.Comment: 26 pages, 1 figure, accepted to WG202