Algorithmic and Hardness Results for the Colorful Components Problems
In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are co...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
06.11.2013
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Subjects | |
Online Access | Get full text |
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Summary: | In this paper we investigate the colorful components framework, motivated by
applications emerging from comparative genomics. The general goal is to remove
a collection of edges from an undirected vertex-colored graph $G$ such that in
the resulting graph $G'$ all the connected components are colorful (i.e., any
two vertices of the same color belong to different connected components). We
want $G'$ to optimize an objective function, the selection of this function
being specific to each problem in the framework.
We analyze three objective functions, and thus, three different problems,
which are believed to be relevant for the biological applications: minimizing
the number of singleton vertices, maximizing the number of edges in the
transitive closure, and minimizing the number of connected components.
Our main result is a polynomial time algorithm for the first problem. This
result disproves the conjecture of Zheng et al. that the problem is $ NP$-hard
(assuming $P \neq NP$). Then, we show that the second problem is $ APX$-hard,
thus proving and strengthening the conjecture of Zheng et al. that the problem
is $ NP$-hard. Finally, we show that the third problem does not admit
polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for
any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 -
\epsilon}$, assuming $ZPP \neq NP$). |
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DOI: | 10.48550/arxiv.1311.1298 |