Robust Popular Matchings

TL;DR

This work investigates robustness in popular matchings by studying whether a matching remains popular across multiple instances of the matching problem under perturbations. It develops a polynomial-time algorithm for RobustPopularMatching when only a single agent changes preferences, via hybrid instances and reductions to PopularEdge, and extends this to RobustDominantMatching. The paper then establishes hardness results for two-agent perturbations (both two same-type agents and two swaps, across sides), revealing a sharp complexity boundary. It also analyzes robustness under reduced availability, deriving a complete-vs-incomplete dichotomy, and discusses robustness within related models such as strong and mixed popularity. Overall, the results delineate when robust popular outcomes are efficiently computable and when they become intractable, guiding future exploration of alternative robustness notions and maximum-size robust solutions.

Abstract

We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent. The same method applies also to dominant matchings, a subclass of maximum-size popular matchings. By contrast, we obtain NP-completeness if two instances differ only by two agents of the same side or by a swap of two adjacent alternatives by two agents. The first hardness result applies to dominant matchings as well. Moreover, we find another complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable. We conclude by discussing related models, such as strong and mixed popularity.

Figures (9)

• Figure 1: Instance $\mathcal{I}_A$ in \ref{['ex:basic']}. The perturbed instance $\mathcal{I}_B$ is obtained by having agent $w_1$ swap their preferences for $f_1$ and $f_3$.
• Figure 2: Illustration of \ref{['thm:twooneside']} for an instance $C_1\wedge C_2\wedge C_3$ with $C_1 = X_1\vee X_2 \vee X_3$, $C_2 = \overline{X}_4\vee \overline{X}_1\vee \overline{X}_2$, and $C_3 = X_5\vee X_4\vee X_1$. Workers are represented by circles and firms by squares. For the sake of clarity, the edges between $v_2$ and vertices of $W\setminus \{ w_1,w_2,t_2,t_3\}$ are omitted and represented by the outgoing dashed edges from $v_2$. These are best for agents of the type $\overline{d}_i^j$ and worst for the others. The numbers on edges denote the rank in the preference lists, where $\ell$ and $\ell-1$ denoting the last second-to-last options, respectively. For the two agents $s_1$ and $v_2$ that changed their preferences in $\mathcal{I}_B$ compared to $\mathcal{I}_A$, we depict their preferences in $\mathcal{I}_A$ on the left in blue and in $\mathcal{I}_B$ on the right in red.
• Figure 3: Illustration of the constructed matching $M$ in \ref{['claim:matching']} for the instance $\mathcal{I}_A$ in the example of \ref{['fig:constrCombined']}. The matching corresponds to the assignment where $X_1$ and $X_4$ are set to False, and $X_2$, $X_3$, and $X_5$ are set to True.
• Figure 4: Auxiliary agents for the forbidden edge $e = \{a,b\}$ of the reduced instances in the proof of \ref{['thm:hardness']}. The preferences in $\mathcal{I}_A$ and $\mathcal{I}_B$ are described in the left and right picture, respectively. The only agent that changes their preferences by a simple swap is agent $a$, as highlighted in red.
• Figure 5: Auxiliary agents for the forced vertex $d$ of the reduced instances in the proof of \ref{['thm:hardness']}. The preferences in $\mathcal{I}_A$ and $\mathcal{I}_B$ are described in the left and right picture, respectively. The only agent that changes their preferences by a simple swap is agent $\ell_d$, as highlighted in red.

Theorems & Definitions (35)

• Theorem 3.1: HuKa11a
• Example 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Corollary 4.3
• Lemma 4.4
• proof
• Lemma 4.5: CsKa17a