Minimum Reload Cost Cycle Cover in Complete Graphs
release_xm3vmczg4fapxp7rq3er2zeztq
by
Yasemin Büyükçolak, Didem Gözüpek, Sibel Özkan
2017
Abstract
The reload cost refers to the cost that occurs along a path on an
edgecolored graph when it traverses an internal vertex between two edges of
different colors. Galbiati et al.[1] introduced the Minimum Reload Cost Cycle
Cover problem, which is to find a set of vertexdisjoint cycles spanning all
vertices with minimum reload cost. They proved that this problem is strongly
NPhard and not approximable within 1/ϵ for any ϵ > 0 even
when the number of colors is 2, the reload costs are symmetric and satisfy the
triangle inequality. In this paper, we study this problem in complete graphs
having equitable or nearly equitable 2edgecolorings, which are edgecolorings
with two colors such that for each vertex v ∈ V(G), c_1(v) c_2(v)
≤ 1 or c_1(v) c_2(v) ≤ 2, respectively, where c_i(v) is the
set of edges with color i that is incident to v. We prove that except
possibly on complete graphs with fewer than 13 vertices, the minimum reload
cost is zero on complete graphs with nearly equitable 2edgecolorings by
proving the existence of a monochromatic cycle cover. Furthermore, we provide a
polynomialtime algorithm that constructs a monochromatic cycle cover in
complete graphs with an equitable 2edgecoloring except possibly in a complete
graph with four vertices. Our algorithm also finds a monochromatic cycle cover
in complete graphs with a nearly equitable 2edgecoloring except some special
cases.
In text/plain
format
Archived Files and Locations
application/pdf 250.0 kB
file_sfmxj7xq3valvm5i73n23nrevm

web.archive.org (webarchive) arxiv.org (repository) 
article
Stage
submitted
Date 20170616
Version
v1
Language
en
^{?}
1706.05225v1
access all versions, variants, and formats of this works (eg, preprints)