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Fair Correlation Clustering Meets Graph Parameters

Johannes Blaha, Robert Ganian, Katharina Gillig, Jonathan S. Højlev, Simon Wietheger

TL;DR

This work investigates the parameterized complexity of Fair Correlation Clustering (FCC), a fairness-aware generalization of Correlation Clustering that uses fairlets to enforce color-ratio constraints within clusters. The authors develop tractable algorithms under several graph-structure parameters: FCC is FPT when parameterized by the vertex cover number, and they provide an XP algorithm in terms of treewidth plus the fairlet size, with a separate FPT result for the case tilde{c} = 2 and an FPT result for treedepth plus tilde{c} via a treedepth-based reduction and ILP. A structural lemma bounds cluster sizes in tree-like graphs, enabling treewidth-based dynamic programming, while vertex-cover-based decomposition reduces to a maximum-weight matching problem to extend pre-clusterings efficiently. Overall, the paper maps two tractable routes for FCC—via vertex cover and via tree-like parameters—and highlights open questions about fixed-parameter tractability when combining treewidth with fairlet size, signaling a nuanced boundary between hardness and tractability in fair clustering problems.

Abstract

We study the generalization of Correlation Clustering which incorporates fairness constraints via the notion of fairlets. The corresponding Fair Correlation Clustering problem has been studied from several perspectives to date, but has so far lacked a detailed analysis from the parameterized complexity paradigm. We close this gap by providing tractability results for the problem under a variety of structural graph parameterizations, including treewidth, treedepth and the vertex cover number; our results lie at the very edge of tractability given the known NP-hardness of the problem on severely restricted inputs.

Fair Correlation Clustering Meets Graph Parameters

TL;DR

This work investigates the parameterized complexity of Fair Correlation Clustering (FCC), a fairness-aware generalization of Correlation Clustering that uses fairlets to enforce color-ratio constraints within clusters. The authors develop tractable algorithms under several graph-structure parameters: FCC is FPT when parameterized by the vertex cover number, and they provide an XP algorithm in terms of treewidth plus the fairlet size, with a separate FPT result for the case tilde{c} = 2 and an FPT result for treedepth plus tilde{c} via a treedepth-based reduction and ILP. A structural lemma bounds cluster sizes in tree-like graphs, enabling treewidth-based dynamic programming, while vertex-cover-based decomposition reduces to a maximum-weight matching problem to extend pre-clusterings efficiently. Overall, the paper maps two tractable routes for FCC—via vertex cover and via tree-like parameters—and highlights open questions about fixed-parameter tractability when combining treewidth with fairlet size, signaling a nuanced boundary between hardness and tractability in fair clustering problems.

Abstract

We study the generalization of Correlation Clustering which incorporates fairness constraints via the notion of fairlets. The corresponding Fair Correlation Clustering problem has been studied from several perspectives to date, but has so far lacked a detailed analysis from the parameterized complexity paradigm. We close this gap by providing tractability results for the problem under a variety of structural graph parameterizations, including treewidth, treedepth and the vertex cover number; our results lie at the very edge of tractability given the known NP-hardness of the problem on severely restricted inputs.
Paper Structure (4 sections, 6 theorems, 3 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Note on Fair Clusters in Tree-like Graphs
  4. Tractability via Vertex Cover Number

Key Result

theorem thmcountertheorem

FCC is in w.r.t. the vertex cover number of $G$.

Figures (1)

  • Figure 1: Left: An input graph; here the fairlet is $\{$red, blue, blue$\}$. Middle: A "fairness-oblivious" minimum-cost correlation clustering (cost 4). Right: A minimum-cost fair correlation clustering (cost 9).

Theorems & Definitions (7)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem