Correlation Clustering with Sherali-Adams
Abstract
Given a complete graph G = (V, E) where each edge is labeled + or −, the Correlation
Clustering problem asks to partition V into clusters to minimize the number of +edges be-
tween different clusters plus the number of −edges within the same cluster. Correlation
Clustering has been used to model a large number of clustering problems in practice, making
it one of the most widely studied clustering formulations. The approximability of Correla-
tion Clustering has been actively investigated [BBC04, CGW05, ACN08], culminating in a
2.06-approximation algorithm by Chawla, Makarychev, Schramm, and Yaroslavtsev [CMSY15].
The state-of-the-art result of Chawla, Makarychev, Schramm, and Yaroslavtsev is obtained by
designing a clever rounding scheme for the standard LP relaxation. Yet, the integrality gap of
this formulation is 2 and so it has remained a major open question to determine whether one
can even breach this bound of 2 for Correlation Clustering.
In this paper, we answer the question affirmatively by showing that there exists a (1.95 +ε)-
approximation algorithm based on O(1/ε^2) rounds of the Sherali-Adams hierarchy. In order to
round a solution to Sherali-Adams, we adapt the correlated rounding originally developed for
CSPs [BRS11, GS11, RT12]. Beyond the traditional triangle-based analysis, we also employ a
global charging scheme that amortizes the total cost of the rounding across different triangles.
We believe our results
could pave the way for applications of LP hierarchies beyond CSPs.