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Approximating graph-constrained max-cut

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Abstract

An instance of the graph-constrained max-cut (\(\mathsf {GCMC}\)) problem consists of (i) an undirected graph \(G=(V,E)\) and (ii) edge-weights \(c:{V\atopwithdelims ()2} \rightarrow \mathbb {R}_+\) on a complete undirected graph. The objective is to find a subset \(S \subseteq V\) of vertices satisfying some graph-based constraint in G that maximizes the weight \(\sum _{u\in S, v\not \in S} c_{uv}\) of edges in the cut \((S,V{\setminus } S)\). The types of graph constraints we can handle include independent set, vertex cover, dominating set and connectivity. Our main results are for the case when G is a graph with bounded treewidth, where we obtain a \(\frac{1}{2}\)-approximation algorithm. Our algorithm uses an LP relaxation based on the Sherali–Adams hierarchy. It can handle any graph constraint for which there is a dynamic program of a specific form. Using known decomposition results, these imply essentially the same approximation ratio for \(\mathsf {GCMC}\) under constraints such as independent set, dominating set and connectivity on a planar graph G.

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Notes

  1. For other polynomial-time dynamic programs, the LP has quasi-polynomial size.

  2. A partition P is said to be satisfied by another partition \(P'\) if every pair of elements in the same part of P also lie in the same part of \(P'\).

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The cleaner proof of Observation 1 presented here was provided by an anonymous reviewer.

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Authors and Affiliations

  1. IOE Department, University of Michigan, Ann Arbor, MI, 48109, USA

    Xiangkun Shen, Jon Lee & Viswanath Nagarajan

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  1. Xiangkun Shen
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Correspondence to Xiangkun Shen.

Additional information

A preliminary announcement of this work appeared as the extended abstract [24].

Research of J. Lee was partially supported by NSF Grant CMMI-1160915 and ONR Grant N00014-14-1-0315.

Research of V. Nagarajan supported in part by a faculty award from Bloomberg Labs.

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Shen, X., Lee, J. & Nagarajan, V. Approximating graph-constrained max-cut. Math. Program. 172, 35–58 (2018). https://doi.org/10.1007/s10107-017-1154-3

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