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An Improved Combinatorial Algorithm for Edge-Colored Clustering in Hypergraphs

Seongjune Han, Nate Veldt

TL;DR

This paper presents the first combinatorial approximation algorithm with an approximation factor better than 2.0, and focuses on clustering such datasets using the NP-hard edge-colored clustering problem, where the goal is to assign colors to nodes in such a way that node colors tend to match edge colors.

Abstract

Many complex systems and datasets are characterized by multiway interactions of different categories, and can be modeled as edge-colored hypergraphs. We focus on clustering such datasets using the NP-hard edge-colored clustering problem, where the goal is to assign colors to nodes in such a way that node colors tend to match edge colors. A key focus in prior work has been to develop approximation algorithms for the problem that are combinatorial and easier to scale. In this paper, we present the first combinatorial approximation algorithm with an approximation factor better than 2.

An Improved Combinatorial Algorithm for Edge-Colored Clustering in Hypergraphs

TL;DR

This paper presents the first combinatorial approximation algorithm with an approximation factor better than 2.0, and focuses on clustering such datasets using the NP-hard edge-colored clustering problem, where the goal is to assign colors to nodes in such a way that node colors tend to match edge colors.

Abstract

Many complex systems and datasets are characterized by multiway interactions of different categories, and can be modeled as edge-colored hypergraphs. We focus on clustering such datasets using the NP-hard edge-colored clustering problem, where the goal is to assign colors to nodes in such a way that node colors tend to match edge colors. A key focus in prior work has been to develop approximation algorithms for the problem that are combinatorial and easier to scale. In this paper, we present the first combinatorial approximation algorithm with an approximation factor better than 2.
Paper Structure (9 sections, 3 theorems, 12 equations, 1 figure, 2 tables, 3 algorithms)

This paper contains 9 sections, 3 theorems, 12 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Deterministic 2-approximation Algorithm
  4. A Faster $(2-2/k)$-Approximation
  5. The LP rounding algorithm
  6. Color Pair Binary Linear Program
  7. Reduction to Flow Network
  8. Experimental Results
  9. Conclusions

Key Result

Theorem 3.1

Algorithm alg:localratioecc is a 2-approximation algorithm for MinECC. It runs in $O(\sum_{v \in V}d_v) = O(\mu)$ time.

Figures (1)

  • Figure 1: In the top left we show a MinECC instance ${H}$, which can be reduced to the $k$-colorable Vertex Cover instance $G$ in the bottom left. On the right, we show the reduced flow network whose minimum $s$-$t$ cut coincides with the optimal solution to $\text{BLP}_\text{CP}$ (see Section \ref{['subsec:reductionalgorithm']}).

Theorems & Definitions (6)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof