3,795 research outputs found
Notes on complexity of packing coloring
A packing -coloring for some integer of a graph is a mapping
such that any two vertices of color
are in distance at least . This concept
is motivated by frequency assignment problems. The \emph{packing chromatic
number} of is the smallest such that there exists a packing
-coloring of .
Fiala and Golovach showed that determining the packing chromatic number for
chordal graphs is \NP-complete for diameter exactly 5. While the problem is
easy to solve for diameter 2, we show \NP-completeness for any diameter at
least 3. Our reduction also shows that the packing chromatic number is hard to
approximate within for any .
In addition, we design an \FPT algorithm for interval graphs of bounded
diameter. This leads us to exploring the problem of finding a partial coloring
that maximizes the number of colored vertices.Comment: 9 pages, 2 figure
Dichotomies properties on computational complexity of S-packing coloring problems
This work establishes the complexity class of several instances of the
S-packing coloring problem: for a graph G, a positive integer k and a non
decreasing list of integers S = (s\_1 , ..., s\_k ), G is S-colorable, if its
vertices can be partitioned into sets S\_i , i = 1,... , k, where each S\_i
being a s\_i -packing (a set of vertices at pairwise distance greater than
s\_i). For a list of three integers, a dichotomy between NP-complete problems
and polynomial time solvable problems is determined for subcubic graphs.
Moreover, for an unfixed size of list, the complexity of the S-packing coloring
problem is determined for several instances of the problem. These properties
are used in order to prove a dichotomy between NP-complete problems and
polynomial time solvable problems for lists of at most four integers
Packing Coloring of Undirected and Oriented Generalized Theta Graphs
The packing chromatic number (G) of an undirected (resp.
oriented) graph G is the smallest integer k such that its set of vertices V (G)
can be partitioned into k disjoint subsets V 1,..., V k, in such a way that
every two distinct vertices in V i are at distance (resp. directed distance)
greater than i in G for every i, 1 i k. The generalized theta graph
{\ell} 1,...,{\ell}p consists in two end-vertices joined by p 2
internally vertex-disjoint paths with respective lengths 1 {\ell} 1
. . . {\ell} p. We prove that the packing chromatic number of any
undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n
3 = |{i / 1 i p, {\ell} i = 3}|, and that both these bounds are
tight. We then characterize undirected generalized theta graphs with packing
chromatic number k for every k 3. We also prove that the packing
chromatic number of any oriented generalized theta graph lies between 2 and 5
and that both these bounds are tight.Comment: Revised version. Accepted for publication in Australas. J. Combi
An Exact Algorithm for the Generalized List -Coloring Problem
The generalized list -coloring is a common generalization of many graph
coloring models, including classical coloring, -labeling, channel
assignment and -coloring. Every vertex from the input graph has a list of
permitted labels. Moreover, every edge has a set of forbidden differences. We
ask for such a labeling of vertices of the input graph with natural numbers, in
which every vertex gets a label from its list of permitted labels and the
difference of labels of the endpoints of each edge does not belong to the set
of forbidden differences of this edge. In this paper we present an exact
algorithm solving this problem, running in time ,
where is the maximum forbidden difference over all edges of the input
graph and is the number of its vertices. Moreover, we show how to improve
this bound if the input graph has some special structure, e.g. a bounded
maximum degree, no big induced stars or a perfect matching
A Landscape Analysis of Constraint Satisfaction Problems
We discuss an analysis of Constraint Satisfaction problems, such as Sphere
Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape.
Several intriguing geometrical properties of the solution space become in this
light familiar in terms of the well-studied ones of rugged (glassy) energy
landscapes. A `benchmark' algorithm naturally suggested by this construction
finds solutions in polynomial time up to a point beyond the `clustering' and in
some cases even the `thermodynamic' transitions. This point has a simple
geometric meaning and can be in principle determined with standard Statistical
Mechanical methods, thus pushing the analytic bound up to which problems are
guaranteed to be easy. We illustrate this for the graph three and four-coloring
problem. For Packing problems the present discussion allows to better
characterize the `J-point', proposed as a systematic definition of Random Close
Packing, and to place it in the context of other theories of glasses.Comment: 17 pages, 69 citations, 12 figure
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