Combinatorial Approximations for Cluster Deletion: Simpler, Faster, and Better

TL;DR

This paper improves approximation guarantees for cluster deletion algorithms, introduces a simple derandomization method, and develops a scalable combinatorial approach for linear programming, enhancing efficiency and practicality.

Contribution

Provides tighter analysis of existing algorithms, introduces a simple derandomization technique, and designs a scalable combinatorial method for linear programming in cluster deletion.

Findings

Approximation guarantees improved from 4 to 3.

Simple greedy derandomization method introduced.

New combinatorial approach for linear programming developed.

Abstract

Cluster deletion is an NP-hard graph clustering objective with applications in computational biology and social network analysis, where the goal is to delete a minimum number of edges to partition a graph into cliques. We first provide a tighter analysis of two previous approximation algorithms, improving their approximation guarantees from 4 to 3. Moreover, we show that both algorithms can be derandomized in a surprisingly simple way, by greedily taking a vertex of maximum degree in an auxiliary graph and forming a cluster around it. One of these algorithms relies on solving a linear program. Our final contribution is to design a new and purely combinatorial approach for doing so that is far more scalable in theory and practice.