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Parameterized algorithms for Max Colorable Induced Subgraph problem on perfect graphs
N. Misra, , A. Rai, V. Raman, S. Saurabh
Published in Springer Verlag
2013
Volume: 8165 LNCS
   
Pages: 370 - 381
Abstract
We address the parameterized complexity of Max Colorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril [IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a nO(q) algorithm on chordal graphs. We first observe that the problem is W[2]-hard parameterized by q, even on split graphs. However, when parameterized by ℓ, the number of vertices in the solution, we give two fixed-parameter tractable algorithms. - The first algorithm runs in time 5.44ℓ (n + #α(G))O(1) where #α(G) is the number of maximal independent sets of the input graph. - The second algorithm runs in time qℓ+o(ℓ) nO(1) Tα where Tα is the time required to find a maximum independent set in any induced subgraph of G. The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in ℓ alone (whenever Tα is a polynomial in n), since q ≤ ℓ for all non-trivial situations. Finally, we show that (under standard complexity-theoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense: (a) On split graphs, we do not expect a polynomial kernel if q is a part of the input. (b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q ≥ 2. © 2013 Springer-Verlag.