Worst- and Average-Case Robustness of Stable Matchings: (Counting) Complexity and Experiments

TL;DR

This paper investigates robustness of stable matchings under preference perturbations in the Stable Marriage setting, introducing worst-case and average-case robustness measures for matchings, pairs, and agents under swaps and deletions. It establishes a mixed complexity landscape: certain decision variants are solvable in polynomial time, while many counting and constructive variants are #P-hard or NP-hard; an FPRAS and Monte Carlo methods are proposed for estimation in practice. Empirically, stability under adversarial swaps is remarkably fragile for typical instances, whereas average-case robustness varies by instance and model, with the blocking-pair proximity heuristic providing a strong, fast predictor. The results highlight that stable pairs are generally more robust than stable matchings, and that robust-and-summed-rank stable matchings can offer practical improvements; the work also introduces Mallows-based experiments and robust-extreme datasets to illuminate these dynamics, with open questions around Pair-Delete-Robustness and parameterized complexity.

Abstract

Focusing on the bipartite Stable Marriage problem, we investigate different robustness measures related to stable matchings. We analyze the computational complexity of computing them and analyze their behavior in extensive experiments on synthetic instances. For instance, we examine whether a stable matching is guaranteed to remain stable if a given number of adversarial swaps in the agent's preferences are performed and the probability of stability when applying swaps uniformly at random. Our results reveal that stable matchings in our synthetic data are highly unrobust to adversarial swaps, whereas the average-case view presents a more nuanced and informative picture.

Figures (28)

• Figure 1: Distribution of the 50%-threshold of men-optimal matching.
• Figure 2: Stability probability of men-optimal matchings in three different instances for varying levels of noise.
• Figure 3: Stability probability of three different matchings in instance sampled from IC model.
• Figure 4: Correlation between 50%-threshold and average number of blocking pairs of men-optimal matching at norm-$\phi=0.1$.
• Figure 5: Distribution of the average $50\%$-threshold of stable pairs in each instance (rounded up to a multiple of $0.02$).

Theorems & Definitions (65)

• Theorem 1
• Proposition 1
• Proposition 1
• Theorem 2
• proof : Proof Sketch
• Theorem 3
• proof : Proof Sketch (Agent)
• Lemma 4
• Proposition 4
• Theorem 5