Advances at the Interface of Combinatorial Optimization and Computations Social Choice: Mathematical Formulations, Structural Decompositions, and Analytical Insights

The rank aggregation problem has ubiquitous applications in operations research, artificial intelligence, computational social choice, and various other fields. Generally, rank aggregation is utilized whenever a set of judges (human or non-human) express their preferences over a set of items, and it is necessary to find a consensus ranking that best represents these preferences collectively. Many real-world instances of this problem involve a very large number of items, include ties, and/or contain partial information, which brings a challenge to decision-makers. This work makes several contributions to overcoming these challenges. Most attention on this problem has focused on an NP-hard distance-based variant known as Kemeny aggregation, for which solution approaches with provable guarantees that can handle difficult large-scale instances remain elusive. Firstly, this work introduces exact and approximate methodologies inspired by the social choice foundations of the problem, namely the Condorcet criterion, to decompose the problem. To deal with instances where exact partitioning does not yield many subsets, it proposes Approximate Condorcet Partitioning, which is a scalable solution technique capable of handling large-scale instances while providing provable guarantees. Secondly, this work delves into the rank aggregation problem under the generalized Kendall-tau distance, which contains Kemeny aggregation as a special case. This new problem provides a robust and highly-flexible framework for handling ties. First, it derives exact and heuristic solution methods for the generalized problem. Second, it introduces a novel social choice property that encloses existing variations of the Condorcet criterion as special cases. Thirdly, this work focuses on top-k list aggregation. Top-k lists are a special form of item orderings wherein out of n total items only a small number of them, k, are explicitly ordered. Top-k lists are being increasingly utilized in various fields including recommendation systems, information retrieval, and machine learning. This work introduces exact and inexact methods for consolidating a collection of heterogeneous top- lists. Furthermore, the strength of the proposed exact formulations is analyzed from a polyhedral point of view. Finally, this work identifies the top-100 U.S. universities by consolidating four prominent university rankings to assess the computational implications of this problem.