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), 415–424] 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.