# Improved Approximation Algorithms for the Maximum Happy Vertices and Edges Problems

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## Abstract

The Maximum Happy Vertices (MHV) problem and the Maximum Happy Edges (MHE) problem are two fundamental problems arising in the study of the homophyly phenomenon in large scale networks. Both of these two problems are NP-hard. Interestingly, the MHE problem is a natural generalization of Multiway Uncut, the complement of the classic Multiway Cut problem. In this paper, we present new approximation algorithms for MHV and MHE based on randomized LP-rounding technique and non-uniform approach. Specifically, we show that MHV can be approximated within \(\frac{1}{\varDelta +1/g(\varDelta )}\), where \(\varDelta \) is the maximum vertex degree and \(g(\varDelta ) = (\sqrt{\varDelta } + \sqrt{\varDelta +1})^2 \varDelta \), and MHE can be approximated within \(\frac{1}{2} + \frac{\sqrt{2}}{4}f(k) \ge 0.8535\), where \(f(k) \ge 1\) is a function of the color number *k*. These results improve over the previous approximation ratios for MHV, MHE as well as Multiway Uncut in the literature.

## Keywords

Maximum Happy Vertices Maximum Happy Edges Approximation algorithm Randomized rounding Network homophyly## Mathematics Subject Classification

68W25 90C27## Notes

### Acknowledgements

We thank the anonymous reviewers for their suggestions which help to improve the presentation of the paper.

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