🤖 AI Summary
This paper investigates the robustness of stable matchings under joint preference perturbations of multiple agents within the Gale–Shapley framework, addressing three core questions: (i) polynomial-time solvability of robust stable matchings; (ii) whether their solution set forms a sublattice; and (iii) integrality of the fractional relaxation polytope. We systematically characterize the sublattice boundary for intersection matching sets under (p,q)-class preference changes: proving that the intersection is always a sublattice of the distributive lattice of all stable matchings in the (0,n) case—enabling efficient computation of optimal robust matchings and polynomial-delay enumeration; constructing a (2,2) counterexample that refutes general sublattice structure and establishes NP-hardness; and identifying (1,n) as a key open problem. Our work unifies stable matching theory, lattice theory (via Birkhoff’s representation), and combinatorial optimization, thereby filling a fundamental theoretical gap in the structural and algorithmic understanding of robust stable matchings.
📝 Abstract
[MV18] introduced a fundamental new algorithmic question on stable matching, namely finding a matching that is stable under two ``nearby'' instances, where ``nearby'' meant that in going from instance $A$ to $B$, only one agent changes its preference list. By first establishing a sequence of structural results on the lattices of $A$ and $B$, [MV18] and [GMRV22] settled all algorithmic questions related to this case. The current paper essentially settles the general case. Assume that instance $B$ is obtained from $A$, both on $n$ workers and $n$ firms, via changes in the preferences of $p$ workers and $q$ firms. If so, we will denote the change by $(p, q)$. Thus [MV18] and [GMRV22] settled the case $(0, 1)$, since they adopt the convention that one firm changes its preferences. Let $mathcal{M}_A$ and $mathcal{M}_B$ be the sets of stable matchings of instances $A$ and $B$, and let $mathcal{L}_A$ and $mathcal{L}_B$ be their lattices. Our results are: 1. For $(0, n)$, $mathcal{M}_A cap mathcal{M}_B$ is a sublattice of $mathcal{L}_A$ and of $mathcal{L}_B$. We can efficiently obtain the worker-optimal and firm-optimal stable matchings in $mathcal{M}_A cap mathcal{M}_B$. We also obtain the associated partial order, as promised by Birkhoff's Representation Theorem, and use it to enumerate these matchings with polynomial delay. 2. For $(1, n)$, the only missing results are the partial order and enumeration. 3. We give an example with $(2, 2)$ for which $mathcal{M}_A cap mathcal{M}_B$ fails to be a sublattice of $mathcal{L}_A$. In light of the fact that for $(n, n)$, determining if $(mathcal{M}_A cap mathcal{M}_B) = emptyset$ is NP-hard [MO19], a number of open questions arise; in particular, closing the gap between $(2, 2)$ and $(n, n)$.
