2,622 research outputs found
Approximation bounds on maximum edge 2-coloring of dense graphs
For a graph and integer , an edge -coloring of is an
assignment of colors to edges of , such that edges incident on a vertex span
at most distinct colors. The maximum edge -coloring problem seeks to
maximize the number of colors in an edge -coloring of a graph . The
problem has been studied in combinatorics in the context of {\em anti-Ramsey}
numbers. Algorithmically, the problem is NP-Hard for and assuming the
unique games conjecture, it cannot be approximated in polynomial time to a
factor less than . The case , is particularly relevant in practice,
and has been well studied from the view point of approximation algorithms. A
-factor algorithm is known for general graphs, and recently a -factor
approximation bound was shown for graphs with perfect matching. The algorithm
(which we refer to as the matching based algorithm) is as follows: "Find a
maximum matching of . Give distinct colors to the edges of . Let
be the connected components that results when M is
removed from G. To all edges of give the th color."
In this paper, we first show that the approximation guarantee of the matching
based algorithm is for graphs with perfect matching
and minimum degree . For , this is better than the approximation guarantee proved in {AAAP}. For triangle free graphs
with perfect matching, we prove that the approximation factor is , which is better than for .Comment: 11pages, 3 figure
The Edge Group Coloring Problem with Applications to Multicast Switching
This paper introduces a natural generalization of the classical edge coloring
problem in graphs that provides a useful abstraction for two well-known
problems in multicast switching. We show that the problem is NP-hard and
evaluate the performance of several approximation algorithms, both analytically
and experimentally. We find that for random -colorable graphs, the number
of colors used by the best algorithms falls within a small constant factor of
, where the constant factor is mainly a function of the ratio of the
number of outputs to inputs. When this ratio is less than 10, the best
algorithms produces solutions that use fewer than colors. In addition,
one of the algorithms studied finds high quality approximate solutions for any
graph with high probability, where the probability of a low quality solution is
a function only of the random choices made by the algorithm
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Cubical coloring -- fractional covering by cuts and semidefinite programming
We introduce a new graph invariant that measures fractional covering of a
graph by cuts. Besides being interesting in its own right, it is useful for
study of homomorphisms and tension-continuous mappings. We study the relations
with chromatic number, bipartite density, and other graph parameters.
We find the value of our parameter for a family of graphs based on
hypercubes. These graphs play for our parameter the role that circular cliques
play for the circular chromatic number. The fact that the defined parameter
attains on these graphs the `correct' value suggests that the definition is a
natural one. In the proof we use the eigenvalue bound for maximum cut and a
recent result of Engstr\"om, F\"arnqvist, Jonsson, and Thapper.
We also provide a polynomial time approximation algorithm based on
semidefinite programming and in particular on vector chromatic number (defined
by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite
programming, J. ACM 45 (1998), no. 2, 246--265]).Comment: 17 page
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