Condorcet Dimension and Pareto Optimality for Matchings and Beyond

TL;DR

This work investigates Condorcet-dimension and Pareto-optimality in one-sided matching and related combinatorial structures. It uncovers a fundamental link showing that any Pareto-optimal set of two (matroid-constrained) matchings is a (weak) Condorcet-winning set, extending to partial orders and arborescences, while revealing that partial orders can force the Condorcet dimension to grow as Θ(√n) (and to Θ(n) with matroid constraints). The paper also proves NP-hardness for deciding fixed-size Condorcet-winning sets and for finding Pareto-optimal matchings under partial orders, highlighting computational barriers in non-strict preference settings. Additionally, for arborescences the Condorcet-dimension bound remains constant (2), complementing the broader understanding of how structure and preference rigidity interact to shape existence and complexity results in these combinatorial elections.

Abstract

We study matching problems in which agents form one side of a bipartite graph and have preferences over objects on the other side. A central solution concept in this setting is popularity: a matching is popular if it is a (weak) Condorcet winner, meaning that no other matching is preferred by a strict majority of agents. It is well known, however, that Condorcet winners need not exist. We therefore turn to a natural and prominent relaxation. A set of matchings is a Condorcet-winning set if, for every competing matching, a majority of agents prefers their favorite matching in the set over the competitor. The Condorcet dimension is the smallest cardinality of a Condorcet-winning set. Our main results reveal a connection between Condorcet-winning sets and Pareto optimality. We show that any Pareto-optimal set of two matchings is, in particular, a Condorcet-winning set. This implication continues to hold when we impose matroid constraints on the set of matched objects, and even when agents' valuations are given as partial orders. The existence picture, however, changes sharply with partial orders. While for weak orders a Pareto-optimal set of two matchings always exists, this is -- surprisingly -- not the case under partial orders. Consequently, although the Condorcet dimension for matchings is 2 under weak orders (even under matroid constraints), this guarantee fails for partial orders: we prove that the Condorcet dimension is $Θ(\sqrt{n})$, and rises further to $Θ(n)$ when matroid constraints are added. On the computational side, we show that, under partial orders, deciding whether there exists a Condorcet -- winning set of a given fixed size is NP-hard. The same holds for deciding the existence of a Pareto-optimal matching, which we believe to be of independent interest. Finally, we also show that the Condorcet dimension for a related problem on arborescences is also 2.

Figures (4)

• Figure 1: The construction of the branching used in the proof of \ref{['thm:warm-up']}. The graph on the left depicts the matching instance, with dark-blue edges belonging to $M_1$, light-blue edges belonging to $M_2$, and red edges belonging to $N$. The resulting branching is depicted on the right.
• Figure 2: Example for the construction of the branching $B$ for matroid-constrained matchings $\mathcal{M} = \{M_1, M_2\}$ and agents $A = \{a_1, \dots, a_8\}$. The agents in $A_+ = \{a_1, a_2, a_5\}$ are colored red, the agents in $A_- = \{a_3, a_4, a_6, a_7\}$ are colored blue, and the remaining agent $a_8$ is color grey. The left and middle figure depict the digraphs $D_{M_1}$ and $D_{M_2}$ respectively, with the arcs on paths $P^{M}_a$ highlighted. Note that $b^{M_1}_{a_1} = a_3$ is the first blue agent encountered when traversing the cycle in $D_{M_1}$ starting from $a_1$, yielding the path $P^{M_1}_{a_1} = \{(a_1, a_2), (a_2, a_3)\}$. Note that $P^{M_1}_{a_2} = \{(a_2, a_3)\} \subseteq P^{M_1}_{a_1}$ and $P^{M_1}_{a_5} = \{(a_5, a_6)\}$. Thus $A^{M_1}_+ = \{a_1, a_5\}$. Similarly, $b^{M_1}_{a_2} = a_4$ with $P^{M_2}_{a_2} = \{(a_2, a_5), (a_5, a_4)\}$, which contains $P^{M_2}_{a_5} = \{(a_5, a_4)\}$. Hence $A^{M_2}_+ = \{a_1, a_2\}$. The figure to the right indicates the resulting matching $B = P^{M_1}_{a_1} \cup P^{M_1}_{a_5} \cup P^{M_2}_{a_1} \cup P^{M_2}_{a_2}$.
• Figure 3: The construction used to prove \ref{['thm:NP-pareto']}. The graph on nodes $V = \{u, v, w\}$ on the left depicts a Vertex Cover instance, for which $\{v\}$ is a vertex cover of size $\ell = 1$. The resulting matching instance is depicted on the right, with $\bar{O}$ containing $|V| - \ell = 2$ nodes. Any agent prefers a solid edge over a dashed edges of the same color, but is indifferent for all other pairs (in particular edges of different colors). The edges marked with grey background form a Pareto-optimal matching.
• Figure 4: Assignment instance showing that Pareto optimal sets need not be Condorcet-winning in the assignment setting. Circles refer to agents and square to objects. The edge patterns indicate the agents' preferences over the objects, where solid is preferred over dashed over dotted. The left image illustrates the set of blue matchings $\mathcal{M}$ and the right image illustrates the competitor assignment $N$ in red. In the right image, agents are colored in accordance to their preferences when comparing $\mathcal{M}$ vs. $N$.

Theorems & Definitions (55)

• theorem 1
• lemma 1
• proof
• proof : Proof of \ref{['thm:warm-up']}
• Claim 2
• proof
• theorem 3
• lemma 2
• corollary 1
• corollary 2