Stable matchings with switching costs

Boris Pittel; Kirill Rudov

Home/Archive/arXiv

arXiv · 2507.14362

Stable matchings with switching costs

Boris Pittel·Kirill Rudov

View original source ↗Read source document ↗Export metadata ↓

Abstract

In a stable matching problem there are two groups of agents, with agents on one side having their individual preferences for agents on another side as a potential match. It is assumed silently that agents can freely and costlessly ``switch" partners. A matching is called stable if no two unmatched agents prefer each other to their matches. Half a century ago, for equinumerous sides, Knuth demonstrated existence of preferences for which there are exponentially many stable matchings, and he posed a problem of evaluating an expected number of stable matchings when the preferences are uniformly random and independent. It was shown later by Pittel that this expectation is quite moderate, asymptotic to $e^{-1}n\log n$, $n$ being the number of agents on each side. The proof used Knuth's integral formula for the expectation based on a classic inclusion-exclusion counting. In later papers by Pittel this and other integral formulas were obtained by generating preference lists via a pair of random matrices, with independently random, $[0,1]$-uniform entries, rather than from a progressively problematic counting approach. The novelty of this paper is that we view the matrix entries as a basis for cardinal utilities that each agent ascribes to the agents on the other side. Relaxing the notion of stability, we declare matching stable if no unmatched pair of agents {\it strongly\/} prefer each other to their partners, with ``strength'' measured by a parameter $\eps>0$, $\eps=0$ corresponding to classic stability. We show that for $\eps$ of order $n^{-1}\log n$ the expected number of $\eps$-stable matchings is polynomially large, but for $\eps\gg n^{-1}\log n$ {\it however slightly\/} the expectation is suddenly super-polynomially large. This ``explosion'' phenomenon for a very small strength parameter $\eps$ continues to hold for imbalanced matchings, regardless of the imbalance size.

Explore related subjects

math.CO math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Boris Pittel, Kirill Rudov. 2025-07-18. Stable matchings with switching costs. https://arxiv.org/abs/2507.14362

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Balanced even cycles in signed graphs:Tur\'an bounds, double covers, and parity obstructions

We study Tur\'an problems for balanced even cycles in simple signed graphs, where signed subgraphs are considered up to switching. For every balanced bipartite signed graph, the signed and ordinary Tur\'an numbers differ by at most a factor of two. Our main structural results concern the underlying graphs that admit a signing in which every $2k$-cycle is unbalanced. We characterize these graphs by the absence of an odd dependence among their $2k$-cycle incidence vectors, give a cohomological formulation, and construct subgraph-minimal obstructions of arbitrarily large order. In particular, there is no finite forbidden-subgraph characterization. We also give an exact closed-walk criterion for cycles in double covers and derive a direct signed breadth-first-search upper bound. As applications, we prove \[ \hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2} \] and study the signed hexagon number $R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$. We characterize the underlying graphs counted by $R_6$ and express it as an extremal problem for ordinary $C_6$-free graphs with a prescribed involution. For every sufficiently large $n$, we construct examples with $\Omega(n^{4/3})$ edges, and we give an equivariant construction attaining the coefficient obtained from the F\"uredi--Naor--Verstra\"ete lower bound by double-cover transfer. Finally, we give $n$-vertex $C_{+10}$-free signed graphs with $\Omega(n^{6/5})$ edges and use octagon examples to illustrate the limitations of theta-freeness as a signing criterion.

math.CO↗

arXiv2026

Fractional DP-colorings of $d$-degenerate locally sparse graphs

Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $\chi_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $\chi_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ \chi_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $\chi_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $\chi_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $\Delta$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.

math.CO↗

arXiv2026

Erd\H{o}s-S\'{o}s for digraphs

It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs. Previously, such tight bounds were not known, even just for directed paths. This can be considered as a directed analog of the recently proved Erd\H{o}s-S\'os conjecture. The result was proved by GPT-6 Astra.

math.CO↗