Deterministic Pivoting Algorithms for Constrained Ranking and Clustering Problems

Abstract

We consider ranking and clustering problems related to the aggregation of inconsistent information, in particular, rank aggregation, (weighted) feedback arc set in tournaments, consensus and correlation clustering, and hierarchical clustering. Ailon et al. [Ailon, N., M. Charikar, A. Newman. 2005. Aggregating inconsistent information: Ranking and clustering. Proc. 37th Annual ACM Sympos. Theory Comput. (STOC '05), 684-693], Ailon and Charikar [Ailon, N., M. Charikar. 2005. Fitting tree metrics: Hierarchical clustering and phylogeny. Proc. 46th Annual IEEE Sympos. Foundations Comput. Sci. (FOCS '05), 73-82], and Ailon [Ailon, N. 2007. Aggregation of partial rankings, p-ratings and top-m lists. Proc. 18th Annual ACM-SIAM Sympos. Discrete Algorithms (SODA '07), [415][416][417][418][419][420][421][422][423][424] proposed randomized constant factor approximation algorithms for these problems, which recursively generate a solution by choosing a random vertex as "pivot" and dividing the remaining vertices into two groups based on the pivot vertex.In this paper, we answer an open question in these works by giving deterministic approximation algorithms for these problems. The analysis of our algorithms is simpler than the analysis of the randomized algorithms. In addition, we consider the problem of finding minimum-cost rankings and clusterings that must obey certain constraints (e.g., an input partial order in the case of ranking problems), which were introduced by Hegde and Jain [Hegde, R., K. Jain. 2006. Personal communication]. We show that the first type of algorithms we propose can also handle these constrained problems. In addition, we show that in the case of a rank aggregation or consensus clustering problem, if the input rankings or clusterings obey the constraints, then we can always ensure that the output of any algorithm obeys the constraints without increasing the objective value of the solution.

Improved Approximation Algorithms for Bipartite Correlation Clustering

“…It starts by solving a Linear Program in order to convert the problem to a non bipartite instance (CC) and then uses the pivoting algorithm [13] to construct a clustering. The algorithm is similar to the one in [11] where nodes from the graph are chosen randomly as 'pivots' or 'centers' and clusters are generated from their neighbor sets.…”

“…The algorithm is similar to the one in [11] where nodes from the graph are chosen randomly as 'pivots' or 'centers' and clusters are generated from their neighbor sets. Arguments from [13] derandomize this choice and give us a deterministic 4-approximation algorithm. This algorithm, unfortunately, becomes impractical for large graphs.…”

“…QuickCluster recursively constructs a clustering simply by iteratively choosing a pivot vertex i at random, forming a cluster C that contains i and all vertices j such that (i, j) ∈ E + , removing them from the graph and repeating. Van Zuylen and Williamson [13] replace the random choice of pivot by a deterministic one, and show conditions under which the resulting algorithm output is a constant factor approximation with respect to the LP objective function. To describe their choice of pivot, we need the notion of a "bad triplet" [11].…”

“…Consider the pairs of vertices on which the output of QuickCluster disagrees with E + and E − , i.e., pairs (i, j) ∈ E − that are in the same cluster, and pairs (i, j) ∈ E + that are not in the same cluster in the output clustering. It is not hard to see that in both cases, there was some call to QuickCluster in which (i, j, k) formed a bad triplet with the pivot vertex k. The pivot chosen by Van Zuylen and Williamson [13] is the pivot that minimizes the ratio of the weight of the edges that are in a bad triplet with the pivot and their LP contribution.…”

“…Van Zuylen and Williamson [13] provided de-randomization for the algorithms presented in [11] with no compromise in the approximation guarantees. Giotis and Guruswami [14] gave a PTAS for the CC case in which the number of clusters is constant.…”