Single-Winner Voting on Matchings
Niclas Boehmer, Jessica Dierking
TL;DR
This work studies single-winner voting on matchings, where candidates are all feasible matchings in a graph and voters have global preferences over entire matchings. It develops a complete complexity landscape across three utility models—affine, one-edge approval, and $\kappa$-missing approval—for three classical solution concepts: social welfare, Pareto optimality, and Condorcet winners. Key findings include a polynomial-time reduction to maximum-weight matching yielding tractable Utilitarian Welfare under affine utilities, while Egalitarian Welfare and verification tasks often become NP-hard or coNP-hard, with sharp complexity jumps when changing $\kappa$ or the solution concept. Restricting the candidate space to maximal matchings yields limited improvements (e.g., XP algorithms for fixed $\kappa$) but largely does not restore tractability, underscoring a fundamental difficulty in aggregating over exponentially many complex outcomes. The results illuminate the limits of applying classic single-winner aggregation concepts to structured, combinatorial domains like matchings, and point to directions for targeted restrictions and parameterized approaches.
Abstract
We introduce a single-winner perspective on voting on matchings, in which voters have preferences over possible matchings in a graph, and the goal is to select a single collectively desirable matching. Unlike in classical matching problems, voters in our model are not part of the graph; instead, they have preferences over the entire matching. In the resulting election, the candidate space consists of all feasible matchings, whose exponential size renders standard algorithms for identifying socially desirable outcomes computationally infeasible. We study whether the computational tractability of finding such outcomes can be regained by exploiting the matching structure of the candidate space. Specifically, we provide a complete complexity landscape for questions concerning the maximization of social welfare, the construction and verification of Pareto optimal outcomes, and the existence and verification of Condorcet winners under one affine and two approval-based utility models. Our results consist of a mix of algorithmic and intractability results, revealing sharp boundaries between tractable and intractable cases, with complexity jumps arising from subtle changes in the utility model or solution concept.