Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A 4-approximation algorithm for min max correlation clustering

Holger Heidrich, Jannik Irmai, Bjoern Andres

TL;DR

This work addresses min max correlation clustering on complete graphs by introducing a combinatorial lower bound (CLB) and deriving a combinatorial $4$-approximation algorithm. It further augments the algorithm with a greedy joining heuristic that yields improvements in solution quality on practical instances. Empirical results show that CLB can be computed much faster than LP-based bounds while achieving competitive or better maximum disagreements compared to state-of-the-art methods, enabling large-scale graphs to be tackled efficiently. The paper also discusses extensions to non-complete and weighted graphs and highlights open questions about combining bounds and improving theoretical guarantees for the heuristic.

Abstract

We introduce a lower bounding technique for the min max correlation clustering problem and, based on this technique, a combinatorial 4-approximation algorithm for complete graphs. This improves upon the previous best known approximation guarantees of 5, using a linear program formulation (Kalhan et al., 2019), and 40, for a combinatorial algorithm (Davies et al., 2023a). We extend this algorithm by a greedy joining heuristic and show empirically that it improves the state of the art in solution quality and runtime on several benchmark datasets.

A 4-approximation algorithm for min max correlation clustering

TL;DR

This work addresses min max correlation clustering on complete graphs by introducing a combinatorial lower bound (CLB) and deriving a combinatorial -approximation algorithm. It further augments the algorithm with a greedy joining heuristic that yields improvements in solution quality on practical instances. Empirical results show that CLB can be computed much faster than LP-based bounds while achieving competitive or better maximum disagreements compared to state-of-the-art methods, enabling large-scale graphs to be tackled efficiently. The paper also discusses extensions to non-complete and weighted graphs and highlights open questions about combining bounds and improving theoretical guarantees for the heuristic.

Abstract

We introduce a lower bounding technique for the min max correlation clustering problem and, based on this technique, a combinatorial 4-approximation algorithm for complete graphs. This improves upon the previous best known approximation guarantees of 5, using a linear program formulation (Kalhan et al., 2019), and 40, for a combinatorial algorithm (Davies et al., 2023a). We extend this algorithm by a greedy joining heuristic and show empirically that it improves the state of the art in solution quality and runtime on several benchmark datasets.
Paper Structure (10 sections, 7 theorems, 8 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 7 theorems, 8 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. MIN MAX CORRELATION CLUSTERING FOR COMPLETE GRAPHS
  4. COMBINATORIAL LOWER BOUND AND 4-APPROXIMATION
  5. Relation to LP bound
  6. Non-complete and weighted graphs
  7. GREEDY JOINING ALGORITHM
  8. EXPERIMENTS
  9. Datasets
  10. CONCLUSION

Key Result

Lemma 1

Let $G=(V, E)$ be a graph. For every partition $\Pi \in P_V$ and every $u, v \in V$:

Figures (2)

  • Figure 1: For the graph depicted in \ref{['fig:combinatorial-bound-better']}, the combinatorial bound ($3$) is stronger than the LP bound ($\tfrac{7}{4}$). For the graph depicted in \ref{['fig:lp-bound-better']}, the LP bound ($\tfrac{5}{4}$) is stronger than the combinatorial bound ($1$). For details, see \ref{['example:combinatorial-bound-better', 'example:lp-bound-better']}.
  • Figure 2: Depicted above are the maximum disagreements of the partitions computed by the $\mathop{\mathrm{DMN}}\limits$ algorithm and our algorithms $\mathcal{A}$, $\mathcal{A}^*$ as well as the lower bounds according to the combinatorial lower bound ($\mathop{\mathrm{CLB}}\limits$) and the LP bound. Depicted below are the runtimes in seconds of all algorithms and both bounding techniques. $f$ is the number of random flips in a graph with 100 nodes (10 cliques of size 10).

Theorems & Definitions (17)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • ...and 7 more