Beyond Max-Cut: λ-extendible properties parameterized above the Poljak–Turzík bound

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Highlights

  • We derive fixed-parameter algorithms for a generalization of above-guarantee Max-Cut.

  • The generalization also captures properties of oriented/edge-labelled graphs.

  • Our results build on and generalize the work of Crowston et al. (ICALP 2012) on Max-Cut.

  • As a corollary we solve an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Abstract

We define strong λ-extendibility as a variant of the notion of λ-extendible properties of graphs (Poljak and Turzík, Discrete Mathematics, 1986). We show that the parameterized APT(Π) problem — given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that HΠ and H has at least λm+1λ2(n1)+k edges — is fixed-parameter tractable (FPT) for all 0<λ<1, for all strongly λ-extendible graph properties Π for which the APT(Π) problem is FPT on graphs which are O(k) vertices away from being a graph in which each block is a clique. Our results hold for properties of oriented graphs and graphs with edge labels, generalize the recent result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards–Erdős bound, and yield FPT algorithms for several graph problems parameterized above lower bounds.

Keywords

Algorithms and data structures
Fixed-parameter tractable algorithms
Above-guarantee parameterization
Max-Cut
λ-extendible properties

Cited by (0)

A preliminary version of this work appeared in the Proceedings of FSTTCS 2012 [1].

1

Supported by the Indo-German Max Planck Center for Computer Science (IMPECS), Bundesministerium für Bildung und Forschung (BMBF) grant 01OA1001.

2

Part of this work was done while visiting the Max-Planck-Institut für Informatik, supported by IMPECS.

3

Part of this work was done while with the Saarland University, supported by the DFG Cluster of Excellence on Multimodal Computing and Interaction and the DFG project DARE (GU 1023/1-2), while at TU Berlin, supported by the DFG project AREG (NI 369/9), and while visiting the Institute of Mathematical Sciences, Chennai, supported by IMPECS.