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Fully Dynamic Adversarially Robust Correlation Clustering in Polylogarithmic Update Time

Vladimir Braverman, Prathamesh Dharangutte, Shreyas Pai, Vihan Shah, Chen Wang

TL;DR

This work addresses dynamic correlation clustering under adaptive adversaries by introducing an adversarially robust algorithm that maintains an $O(1)$-approximation with polylogarithmic amortized update time. Central to the approach is a sparse-dense decomposition (SDD) that is updated locally in neighborhoods, enabling fast updates while preserving clustering quality. The authors prove amortized time bounds and approximation guarantees, and validate the method with experiments on synthetic SBMs and real networks, showing stable performance and competitive costs. The sparse-dense decomposition technique itself is presented as of independent interest, with potential applications beyond correlation clustering in dynamic graphs.

Abstract

We study the dynamic correlation clustering problem with $\textit{adaptive}$ edge label flips. In correlation clustering, we are given a $n$-vertex complete graph whose edges are labeled either $(+)$ or $(-)$, and the goal is to minimize the total number of $(+)$ edges between clusters and the number of $(-)$ edges within clusters. We consider the dynamic setting with adversarial robustness, in which the $\textit{adaptive}$ adversary could flip the label of an edge based on the current output of the algorithm. Our main result is a randomized algorithm that always maintains an $O(1)$-approximation to the optimal correlation clustering with $O(\log^{2}{n})$ amortized update time. Prior to our work, no algorithm with $O(1)$-approximation and $\text{polylog}{(n)}$ update time for the adversarially robust setting was known. We further validate our theoretical results with experiments on synthetic and real-world datasets with competitive empirical performances. Our main technical ingredient is an algorithm that maintains $\textit{sparse-dense decomposition}$ with $\text{polylog}{(n)}$ update time, which could be of independent interest.

Fully Dynamic Adversarially Robust Correlation Clustering in Polylogarithmic Update Time

TL;DR

This work addresses dynamic correlation clustering under adaptive adversaries by introducing an adversarially robust algorithm that maintains an -approximation with polylogarithmic amortized update time. Central to the approach is a sparse-dense decomposition (SDD) that is updated locally in neighborhoods, enabling fast updates while preserving clustering quality. The authors prove amortized time bounds and approximation guarantees, and validate the method with experiments on synthetic SBMs and real networks, showing stable performance and competitive costs. The sparse-dense decomposition technique itself is presented as of independent interest, with potential applications beyond correlation clustering in dynamic graphs.

Abstract

We study the dynamic correlation clustering problem with edge label flips. In correlation clustering, we are given a -vertex complete graph whose edges are labeled either or , and the goal is to minimize the total number of edges between clusters and the number of edges within clusters. We consider the dynamic setting with adversarial robustness, in which the adversary could flip the label of an edge based on the current output of the algorithm. Our main result is a randomized algorithm that always maintains an -approximation to the optimal correlation clustering with amortized update time. Prior to our work, no algorithm with -approximation and update time for the adversarially robust setting was known. We further validate our theoretical results with experiments on synthetic and real-world datasets with competitive empirical performances. Our main technical ingredient is an algorithm that maintains with update time, which could be of independent interest.

Paper Structure

This paper contains 30 sections, 17 theorems, 13 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Comparison with Assadi022.
  4. Comparison with Cohen-AddadLMP24.
  5. Insertion, deletion, edges, and non-edges.
  6. Experiments.
  7. Technical overview
  8. Additional Works on Correlation Clustering
  9. Concurrent and independent work on dynamic sparse-dense decomposition.
  10. Open problems.
  11. Preliminaries
  12. Notation.
  13. The sparse-dense decomposition.
  14. Algorithms for sparse-dense decomposition.
  15. Our Main Algorithm
  16. ...and 15 more sections

Key Result

Theorem 1

There is an adversarially robust algorithm that given a fully dynamic labeled complete graph $G=(V, E^+ \cup E^-)$ such that the edge labels are adaptively flipped, maintains an implicit representation of an $O(1)$-approximation to the optimal correlation clustering disagreement cost in $\mathop{\ma

Figures (8)

  • Figure 1: An illustration of the overall update strategy and the discrepancy between the local and global sparse-dense decomposition. In \ref{['fig:overall-strategy']}, we run algorithm updates on $G[N[u]]$ once the number of insertions and deletions on $u$ has exceeded $\varepsilon\cdot \deg(u)$. In \ref{['fig:local-decomp-difficulty']}, note that vertex $v_1$ belongs to an almost-clique in $G[N[u]]$, but is sparse globally. $v_2$ is a sparse vertex in $G[N[u]]$, but it belongs to another almost-clique.
  • Figure 2: An illustration of the role of \ref{['alg:AC-dense-test']} and \ref{['lem:dense-merge-test']}. The shaded vertex $v$ should be added to $K$; however, local updates cannot capture this due to the fact that $v$ is not a neighbor of $u$. \ref{['alg:AC-dense-test']} samples the dotted vertices, and recognize their common neighbor, which are the vertices should be added to $K$.
  • Figure 3: An illustration of the analysis in \ref{['lem:not-update-AC-guarantee']} and \ref{['lem:not-update-sparse-guarantee']}. Red solid edges are the insertions and blue dotted edges are the deletions since the almost-clique and sparse vertices are formed. \ref{['fig:without-update-AC']}: there could not be too many edge updates that make the almost-clique sparse since otherwise either the vertex itself becomes sparse, or the entire almost-clique is dismantled. \ref{['fig:without-update-sparse']}: initially all vertices in $S(v)$ is sparse. If sufficiently many of them become not sparse, an update that leads to almost-clique $K$; and by \ref{['alg:AC-dense-test']}, $v$ would have joined $K$.
  • Figure 4: An illustration of the role of \ref{['lem:AC-no-false']}. Apart from the almost-clique that is formed locally, vertices like $v_1$ that are "dense" w.r.t. $K$ will be added by \ref{['alg:AC-dense-test']} (see also \ref{['fig:add-v-to-ac']}). Furthermore, vertices like $v_2$ will be removed if it is sparse (see also \ref{['fig:local-AC-check']}).
  • Figure 5: An illustration of the case analysis in \ref{['lem:sparse-no-false']}. \ref{['fig:sparse-vertex-update-case-a']}: if a vertex $w\in K$ is initially somehow sparse, then $w$ could not be made very dense since otherwise, an algorithm update would happen. \ref{['fig:sparse-vertex-update-case-b']}: if a vertex $w\in K$ is initially very dense, then the additional insertions that makes other vertices in $K$ sparse will also make $w$ sparse.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1: Adjacency list model for dynamic edge label flips
  • Theorem 1
  • Definition 2: Assadi022
  • Proposition 2.1: Assadi022
  • Definition 3: cf. Assadi022AssadiSW23
  • Proposition 2.2: cf. Assadi022AssadiSW23
  • Theorem 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 26 more