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Simultaneously Approximating All $\ell_p$-norms in Correlation Clustering

Sami Davies, Benjamin Moseley, Heather Newman

TL;DR

This paper gives a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)-approximate for all $\ell_p$-norms of the disagreement vector; in other words, a combinatorial $O(1)-approximation of the all-norms objective for correlation clustering.

Abstract

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement vector; in other words, a combinatorial $O(1)$-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the $\ell_2$-norm objective, and more generally the first combinatorial algorithm for the $\ell_p$-norm objective when $1 < p < \infty$. It is also faster than all previous algorithms that minimize the $\ell_p$-norm of the disagreement vector, with run-time $O(n^ω)$, where $O(n^ω)$ is the time for matrix multiplication on $n \times n$ matrices. When the maximum positive degree in the graph is at most $Δ$, this can be improved to a run-time of $O(nΔ^2 \log n)$.

Simultaneously Approximating All $\ell_p$-norms in Correlation Clustering

TL;DR

This paper gives a combinatorial algorithm that returns a single clustering solution that is simultaneously \ell_pO(1)-approximation of the all-norms objective for correlation clustering.

Abstract

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously -approximate for all -norms of the disagreement vector; in other words, a combinatorial -approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the -norm objective, and more generally the first combinatorial algorithm for the -norm objective when . It is also faster than all previous algorithms that minimize the -norm of the disagreement vector, with run-time , where is the time for matrix multiplication on matrices. When the maximum positive degree in the graph is at most , this can be improved to a run-time of .
Paper Structure (28 sections, 15 theorems, 80 equations, 2 figures, 1 algorithm)

This paper contains 28 sections, 15 theorems, 80 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Related work
  4. Preliminaries
  5. Notation
  6. Summary of work by Kalhan, Makarychev, and Zhou
  7. Summary of work by Davies, Moseley, and Newman
  8. Technical overview
  9. A Special Case: Regular Graphs
  10. Proof of Theorem \ref{['thm: main-acm']}
  11. Triangle inequality
  12. Bounding the fractional cost of $\ell_p$-norms
  13. Fractional cost of positive edges in $\ell_p$-norms
  14. Case 1:
  15. Case 2:
  16. ...and 13 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be an instance of unweighted, complete correlation clustering on $|V|=n$ nodes. There exists a combinatorial algorithm returning a single clustering that is simultaneously an $O(1)$-approximationNote this is independent of $p$. for all $\ell_p$-norm objectives, for all $p\in \mathbb{R}

Figures (2)

  • Figure 1: Two clusterings of the star graph, which has one node ($w$) with positive edges to all nodes, and the rest of the edges negative. Left: Clustering assigns all nodes to one (blue) cluster, and is (almost) optimal for the $\ell_\infty$-norm with cost $\Theta(n)$. Right: Clustering assigns all nodes to different clusters and is (almost) optimal for the $\ell_1$-norm with cost $\Theta(n)$. The left solution is terrible for the $\ell_1$-norm, as the negative clique has $\Theta(n^2)$ edges that are disagreements.
  • Figure 2: Left: Case 2a(i). For $v \in |N_u^- \cap C(u)|$, $(u,v)$ is a disagreement. Right: Case 2a(ii). For $w \in |N_u^+ \cap C(u)|$ and $v \in N_w^+ \cap R_1'(u)$, $(w,v)$ is a disagreement.

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 1
  • Definition 1
  • Definition 2: DMN23
  • Definition 3
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Lemma 1: Triangle Inequality
  • ...and 24 more