Elsevier

Theoretical Computer Science

Volume 749, 21 November 2018, Pages 47-58
Theoretical Computer Science

Approximation and hardness results for the Max k-Uncut problem

https://doi.org/10.1016/j.tcs.2017.09.003Get rights and content
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Highlights

  • Max k-Uncut arises from the study of homophily of networks.

  • Max k-Uncut is the complement of the classic Min k-Cut problem.

  • Max k-Uncut is equivalent to Densest k-Subgraph in terms of approximability.

Abstract

In the study of the homophily law of large scale complex networks, we get a combinatorial optimization problem which we call the Max k-Uncut problem. Given an n-vertex undirected graph G=(V,E) with nonnegative weights {we|eE} defined on edges, and a positive integer k, the Max k-Uncut problem asks to find a partition {V1,V2,,Vk} of V such that the total weight of edges that are not cut is maximized. Intuitively, an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition. Interestingly, the Max k-Uncut problem is just the complement of the classic Min k-Cut problem. For Max k-Uncut, we present a randomized (1kn)2-approximation algorithm, a greedy (12(k1)n)-approximation algorithm, and an Ω(12α)-approximation algorithm by reducing it to Densest k-Subgraph, where α is the approximation ratio of the Densest k-Subgraph problem. More importantly, we show that Max k-Uncut and Densest k-Subgraph are in fact equivalent in approximability up to a factor of 2. We also prove an approximation hardness result for Max k-Uncut under the assumption PNP.

Keywords

Max k-Uncut
Densest k-Subgraph
Homophily
Approximation algorithm
Computational complexity

Cited by (0)

A preliminary version of this paper appeared in the Proceedings of the 10th International Conference of Combinatorial Optimization and Applications (COCOA 2016) [25].

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Part of the author's work was done when he was visiting at the University of California, Riverside, USA, and at Beijing University of Technology, China.