Abstract
In this paper we study the minimum rainbow subgraph problem, motivated by applications in bioinformatics. The input of the problem consists of an undirected graph where each edge is coloured with one of the p possible colors. The goal is to find a subgraph of minimum order (i.e. minimum number of vertices) which has precisely one edge from each color class.
In this paper we show a \(\max(\sqrt{2p}, \min_q(q + \frac{\Delta}{e^{p q^2/\Delta n}}))\)-approximation algorithm using LP rounding, where Δ is the maximum degree in the input graph. In particular, this is a \(\max(\sqrt{2n}, \sqrt{2\Delta\ln{\Delta}})\)-approximation algorithm. On the other hand we prove that there exists a constant c such that the minimum rainbow subgraph problem does not have a cln Δ-approximation, unless NP ⊆ TIME(n O(loglogn)).
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Popa, A. (2012). Approximating the Rainbow – Better Lower and Upper Bounds. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_17
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DOI: https://doi.org/10.1007/978-3-642-32241-9_17
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