A unifying approach to splitting-off

Abstract

We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edge-connectivity augmentation of graphs, global arc-connectivity augmentation of mixed graphs with undirected edges, or the node-to-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can get stuck. This characterization enables us to give very simple proofs for the results mentioned above. Finally we apply our observations on generalizations of the above problems: we consider three connectivity augmentation problems in hypergraphs where the objective is to use hyperedges of minimum total size without increasing the rank. The first is local edge-connectivity augmentation of undirected hypergraphs. The second is global arc-connectivity augmentation of mixed hypergraphs with undirected hyperedges. The third is a hypergraphic generalization of the node-to-area connectivity augmentation problem. We show that a greedy approach (almost) works in these cases.

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Correspondence to Attila Bernáth.

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An extended abstract version of this paper has appeared in the proceedings of IPCO 2008, pages 401–415

Supported by OTKA grants K60802 and TS 049788.

Supported by grants OTKA K60802 and OMFB-01608/2006.

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Bernáth, A., Király, T. A unifying approach to splitting-off. Combinatorica 32, 373–401 (2012). https://doi.org/10.1007/s00493-012-2548-8

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Mathematics Subject Classification (2000)

  • 90C27. 05C40
  • 05C85