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 (certain type of) dynamic program that exactly optimizes linear objectives.
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 (more generally for bounded-genus or excluded-minor graphs).
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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Notes
- 1.
For other polynomial-time dynamic programs, the LP has quasi-polynomial size.
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Lee, J., Nagarajan, V., Shen, X. (2016). Max-Cut Under Graph Constraints. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_5
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