arxiv: 2604.27823 · v1 · submitted 2026-04-30 · 💻 cs.GT

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Maximally Diverse Stable Matchings: Optimizing Arbitrary Institutional Objectives

Gergely Cs\'aji , Zhaohong Sun

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Pith reviewed 2026-05-07 07:47 UTC · model grok-4.3

classification 💻 cs.GT

keywords stable matchingsdiversity quotasinstitutional objectivespolynomial-time algorithmsutilitarian objectiveegalitarian objectiveprice of stabilitysibling co-assignment

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The pith

Stable matchings can efficiently optimize any polynomial-time institutional objective.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that within the set of stable matchings, one can efficiently find the one that best achieves arbitrary institutional goals such as diversity or family co-location. The key is a reduction that converts any computable set function measuring how good an institution's assigned students are into simple linear weights on possible assignments. This allows existing stable matching solvers to optimize the global objective without compromising stability. A reader should care because it means matching systems no longer have to choose between keeping matches stable and meeting complex diversity targets; both can be done in polynomial time. It also provides a way to calculate the cost of requiring stability when pure diversity maximization would do better.

Core claim

We introduce a general, polynomial-time algorithmic framework to optimize arbitrary institutional (or even two-sided) objectives within the set of stable matchings. We prove that for any polynomial-time computable set functions g_i evaluating the assigned students at institutions i in I, a stable matching minimizing either the utilitarian objective sum g_i or the egalitarian objective max g_i can be found efficiently. Our approach leverages the structural properties of stable matchings, mapping arbitrary set functions to linear edge weights. We apply this to solve open problems including stable matchings that minimally violate overlapping diversity quotas and that maximize sibling co-allocat

What carries the argument

The mapping of arbitrary polynomial-time computable set functions g_i to linear edge weights on the assignment graph, which preserves the optimality of minimum-cost stable matchings for the original objective.

If this is right

Stable matchings minimizing total violations of overlapping diversity quotas can be computed efficiently.
Stable matchings minimizing the maximum violation of diversity quotas can be computed efficiently.
Stable matchings maximizing the number of sibling families at the same institution can be found efficiently.
The gap between the maximally diverse matching and the maximally diverse stable matching can be quantified for these objectives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This weight-mapping technique could extend to optimize stable outcomes in related markets such as kidney exchange or housing allocation.
Clearinghouses could apply the framework to balance multiple goals like diversity and stability using a single algorithm.
Testing the method on real school choice datasets would show how large the price of stability typically is for diversity goals.
The approach relies on polynomial-time computability of the objectives, so non-computable or NP-hard goals would need separate handling.

Load-bearing premise

That every polynomial-time computable set function on assigned students can be converted into linear edge weights such that the minimum-cost stable matching solves the original optimization problem exactly.

What would settle it

A small instance with known optimal stable matching for a simple poly-time diversity set function g_i, where applying the edge-weight mapping and running minimum-cost stable matching yields a different outcome.

read the original abstract

Stable matching theory is the foundation of centralized clearinghouses worldwide, from school choice programs to medical residency allocations. However, incorporating complex distributional goals-such as multi-dimensional diversity quotas or sibling co-assignment guarantees-often compromises stability or renders the problem computationally intractable. The existing literature typically addresses this tension by weakening stability to accommodate distributional constraints. In contrast, the reverse question remains largely unexplored: if we restrict attention to stable matchings, to what extent can such distributional objectives be achieved? In this paper, we resolve this tension by introducing a general, polynomial-time algorithmic framework to optimize arbitrary institutional (or even two-sided) objectives within the set of stable matchings. We prove that for any polynomial-time computable set functions $g_i$ evaluating the assigned students at institutions $i \in I$, a stable matching minimizing either the utilitarian objective $\sum_{i\in I} g_i$ or the egalitarian objective $\max_{i\in I} g_i$ can be found efficiently. Our approach leverages the structural properties of stable matchings, mapping arbitrary set functions to linear edge weights. We apply this theorem to efficiently solve major open practical problems: finding stable matchings that minimally violate overlapping diversity quotas (under both total and maximum violations) and maximizing the number of sibling families assigned to the same institution. Even when the distributional objective is prioritized, our algorithm helps to quantify the ``price of stability'', i.e., the gap between the maximally diverse matching and the maximally diverse stable matching.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's central claim that arbitrary poly-time set functions can be mapped to linear weights for exact optimization over stable matchings does not hold.

read the letter

The main thing to know is that the general theorem does not work. The authors say any poly-time computable g_i can be turned into edge weights so that a standard stable matching algorithm finds the stable matching minimizing the sum or the max of the g_i values. Linear weights produce strictly additive costs, and nothing in the lattice structure changes that for non-additive functions such as coverage, parity, or quadratic terms. The stress-test examples look like they break the reduction, and the abstract's proof sketch does not explain a workaround.

Referee Report

1 major / 0 minor

Summary. The paper claims a general polynomial-time algorithmic framework for optimizing arbitrary institutional objectives within stable matchings. For any polynomial-time computable set functions g_i, it asserts that a stable matching minimizing the utilitarian objective sum g_i or the egalitarian objective max g_i can be found efficiently by mapping the set functions to linear edge weights and applying standard stable matching algorithms (e.g., min-cost variants). The approach is applied to minimizing violations of overlapping diversity quotas and maximizing sibling co-assignments, while also quantifying the price of stability.

Significance. If the central mapping and optimization result hold, the work would be a significant contribution to stable matching theory. It would enable efficient optimization of complex, potentially non-linear distributional goals (such as multi-dimensional diversity or family constraints) strictly within the stable matchings, without weakening stability or incurring intractability. This directly addresses open practical problems in school choice and residency matching and provides a tool to measure the trade-off between stability and diversity objectives.

major comments (1)

Abstract and §3 (main theorem and proof sketch): The claimed mapping of arbitrary polynomial-time computable set functions g_i to linear edge weights w(s,i) such that a min-cost (or min-max) stable matching under w exactly optimizes sum g_i or max g_i cannot hold in general. Linear edge-weight objectives are strictly additive per institution: the contribution of institution i is exactly sum_{s assigned to i} w(s,i). This equals g_i(S_i) for every possible assignment S_i only when g_i itself is additive. The paper states the result for arbitrary g_i, but non-additive functions (e.g., coverage functions g_i(S)=1 iff S intersects every group, parity |S| mod 2, or quadratic |S choose 2|) cannot be represented this way. The lattice structure of stable matchings permits efficient optimization of linear objectives via modified Gale-Shapley or min-cost algorithms, but supplies no such guarantee,

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for raising this important point about the scope of our main result. We agree that the original presentation overstated the generality of the mapping and will revise the manuscript to correct this. The core algorithmic framework remains valid for the class of objectives that admit a polynomial-time mapping to linear edge weights, which includes the practical applications developed in the paper.

read point-by-point responses

Referee: Abstract and §3 (main theorem and proof sketch): The claimed mapping of arbitrary polynomial-time computable set functions g_i to linear edge weights w(s,i) such that a min-cost (or min-max) stable matching under w exactly optimizes sum g_i or max g_i cannot hold in general. Linear edge-weight objectives are strictly additive per institution: the contribution of institution i is exactly sum_{s assigned to i} w(s,i). This equals g_i(S_i) for every possible assignment S_i only when g_i itself is additive. The paper states the result for arbitrary g_i, but non-additive functions (e.g., coverage functions g_i(S)=1 iff S intersects every group, parity |S| mod 2, or quadratic |S choose 2|) cannot be represented this way. The lattice structure of stable matchings permits efficient optimization of linear objectives via modified Gale-Shapley or min-cost algorithms, but supplies no such guarantee,

Authors: We thank the referee for this observation, which correctly notes that fixed linear edge weights can only represent additive objectives. Our claimed framework applies specifically to institutional objectives g_i that admit a polynomial-time computable mapping to edge weights w(s,i) such that the minimum-cost stable matching under these weights optimizes the utilitarian or egalitarian objective. For completely arbitrary non-additive g_i (such as the parity or quadratic examples), no such fixed-weight representation exists in general, and the problem is typically intractable. We will revise the abstract, introduction, and §3 to: (i) restrict the theorem statement to objectives for which such a mapping exists and can be computed efficiently; (ii) provide a complete, self-contained proof detailing the weight-construction procedure and the invocation of the min-cost stable matching algorithm; and (iii) explicitly state the limitations for non-representable functions. The two concrete applications in the paper—minimizing violations of overlapping diversity quotas and maximizing sibling co-assignments—can be encoded via per-student penalties or bonuses and therefore fall inside the representable class. These changes will make the contribution precise without diminishing its practical significance for school choice and residency matching. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external structural properties.

full rationale

The paper claims a polynomial-time result by constructing a mapping from arbitrary poly-time computable set functions g_i to linear edge weights, then invoking the known lattice structure and algorithms (e.g., variants of Gale-Shapley or min-cost stable matching) that optimize linear objectives over stable matchings. This mapping and the subsequent optimization step are presented as part of an explicit proof rather than a self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The structural properties invoked are standard results from prior literature on stable matchings and are not justified solely by the authors' own prior work in a way that reduces the central claim to an unverified self-reference. No equations or steps in the provided description equate the output objective to the input by construction. The derivation is therefore self-contained against external benchmarks in matching theory, even if the validity of the general mapping for non-additive g_i remains a separate correctness question.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard definition of stability in bipartite markets and the assumption that the objective functions are polynomial-time computable; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math Stability is defined via the absence of blocking pairs in the standard Gale-Shapley sense.
The paper builds directly on classical stable matching theory to preserve the lattice structure.
domain assumption The set functions g_i are polynomial-time computable.
Required for the reduction to linear weights to remain efficient.

pith-pipeline@v0.9.0 · 5565 in / 1270 out tokens · 59055 ms · 2026-05-07T07:47:07.166226+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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