Approximating the Rainbow – Better Lower and Upper Bounds

  • Alexandru Popa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


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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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexandru Popa
    • 1
  1. 1.Department of Communications & NetworkingAalto University School of Electrical EngineeringAaltoFinland

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