Improved Approximation Algorithms for Bipartite Correlation Clustering

Abstract

Abstract. In this work we study the problem of Bipartite Correlation Clustering (BCC), a natural bipartite counterpart of the well studied Correlation Clustering (CC) problem. Given a bipartite graph, the objective of BCC is to generate a set of vertex-disjoint bi-cliques (clusters) which minimizes the symmetric difference to it. The best known approximation algorithm for BCC due to Amit (2004) guarantees an 11-approximation ratio. 4 In this paper we present two algorithms. The first is an improved 4-approximation algorithm. However, like the previous approximation algorithm, it requires solving a large convex problem which becomes prohibitive even for modestly sized tasks. The second algorithm, and our main contribution, is a simple randomized combinatorial algorithm. It also achieves an expected 4-approximation factor, it is trivial to implement and highly scalable. The analysis extends a method developed by Ailon, Charikar and Newman in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for CC. For analyzing our algorithm for BCC, considerably more sophisticated arguments are required in order to take advantage of the bipartite structure. Whether it is possible to achieve (or beat) the 4-approximation factor using a scalable and deterministic algorithm remains an open problem.

Near Optimal LP Rounding Algorithm for CorrelationClustering on Complete and Complete k-partite Graphs

“…Correlation clustering on complete bipartite graphs was first studied by Amit [5], who presents an 11-approximation. This was subsequently improved to a 4-approximation by Ailon et al [2]. For complete graphs with weights satisfying triangle inequalities a 3-approximation was obtained by Gionis et al [17] and a 2-approximation by Ailon et al [3].…”

Min-Max Correlation Clustering via MultiCut

“…Let P i = S i \ (∪ j<i S j ). 2 Step 2: while There is a set P i such that δ(P i ) > 2B do 3 Set P i = S i and for all j = i, set P j = P j \ S i ;…”

“…Let P i = S i \ (∪ j<i S j ). 2 Step 2: while There is a set P i such that δ(P i ) > 2B do 3 Set P i = S i and for all j = i, set P j = P j \ S i ; 4 Step 3: Let B ′ = max{ Set P i = P i ∪ P j and P j = ∅. 7 Step 4: For each terminal-part P i , combine at most 2 min(2T, n) · ∆ non-terminal parts with it.…”

“…In addition, correlation clustering captures some fundamental graph cut problems including min s-t cut, multiway cut and multicut. Correlation clustering has been studied extensively for more than a decade [2,3,7,8,11]. Most of the papers have focused on a global min-sum objective function, i.e.…”

The Feedback Arc Set Problem with Triangle Inequality Is a Vertex Cover Problem

“…Important ordering problems can be seen as special cases of MinFAS with restrictions on the weighting function. Examples of this kind are provided by ranking problems related to the aggregation of inconsistent information, that have recently received a lot of attention [2,3,4,19,30,31]. Several of these problems can be modeled as (constrained) MinFAS with weights satisfying either triangle inequalities (i.e., for any triple i, j, k, w (i,j) + w (j,k) ≥ w (i,k) ), or probability constraints (i.e., for any pair i, j, w (i,j) + w (j,i) = 1), or both.…”