Abstract

Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online algorithm based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and a strict competitive ratio at most 22.641. We also show that no deterministic algorithm can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is n-omega (1), where n is the number of vertices in the graph.

Highlights

  • The correlation clustering problem and its different variants have been extensively studied over the past decades; see e.g. [1,5,11]

  • The instance of correlation clustering consists of a graph whose vertices represent some objects and edges represent their similarity

  • We focus on the online variant of clique clustering, where the input graph G is not known in advance

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Summary

Introduction

The correlation clustering problem and its different variants have been extensively studied over the past decades; see e.g. [1,5,11]. With only limited information about the input sequence and the restrictions on allowed operations, an online clique clustering algorithm cannot be guaranteed to always compute an optimal solution. Vertices arrive one at a time and clusters need to be built incrementally They prove that for minimizing disagreements the optimal competitive ratio is (n), and that it is achieved by a simple greedy algorithm. We investigate the online clique clustering problem and provide upper and lower bounds for the competitive ratios for its maximization and minimization versions, that we denote MaxCC and MinCC, respectively. We show that the competitive ratio of the greedy algorithm is n − 2, matching this lower bound

Preliminaries
Online Maximum Clique Clustering
The Greedy Algorithm for Online MAXCC
A Constant Competitive Algorithm for MAXCC
Algorithm OCC
Asymptotic Analysis of OCC
Strict Competitive Ratio
A Lower Bound for Algorithm OCC
A Lower Bound of 6 for MAXCC
Online MINCC Clustering
A Lower Bound for Online MINCC Clustering
The Greedy Algorithm for Online MINCC Clustering
Full Text
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