Optimization-Based Many-to-One Assignment

• The paper’s main contribution is the integration of a multi-objective optimization framework that relaxes strict stability to improve cost, matching rates, and overall assignment efficiency.
• It introduces an inverse optimization technique to encode and penalize blocking pairs, enabling scalable solutions beyond traditional Deferred-Acceptance methods.
• Empirical evaluations demonstrate that controlled instability leads to significant gains, such as reduced transportation costs and increased matches for couples in assignment markets.

An optimization-based many-to-one assignment algorithm refers to any approach that formulates and solves the problem of assigning multiple agents (e.g., students, jobs, passengers) to a set of resources or programs (e.g., schools, machines, hospitals), where each resource may be assigned to one or more agents. Recent research has advanced these algorithms beyond classical methods by integrating explicit optimization objectives that manage multiple trade-offs—most notably, relaxing the requirement of stability (i.e., permitting a small, controlled number of blocking pairs) in order to improve other assignment targets, such as minimizing transportation costs or increasing priority satisfaction in real-world matching markets (Bertsimas et al., 17 Sep 2025).

1. Problem Definition and Motivation

Many-to-one assignment markets arise in diverse domains such as school admissions, residency matching, labor allocation, and resource scheduling. The canonical Deferred-Acceptance (DA) algorithm is widely used in these contexts due to its guarantee of producing stable matchings—no agent-resource pair (blocking pair) is motivated to deviate given their reported preferences and priorities. However, strict stability can impose significant constraints:

• It may prevent the system from optimizing other important objectives, for example, total cost (such as transportation or commute distance), maximizing preference satisfaction, or matching couples in medical residency.
• Stability potentially limits the number of assignments that can be made in the first round, leading to unfilled positions even when program capacity remains.
• DA outputs are insensitive to Pareto-improving relaxations that could reduce overall system inefficiency if stability violations (blocking pairs) are acceptably small.

The optimization-based paradigm seeks to address these trade-offs by formally incorporating the instability metric—number or weighted sum of blocking pairs—directly into the objective, optimizing it jointly with policy-relevant targets.

2. Multi-Objective Formulation Incorporating Stability

The core innovation of the algorithm is the explicit multi-objective optimization problem:

• Let $x_{ij}$ be the binary assignment of agent $i$ to resource $j$.
• Let $b_{ij}^*$ be a cost coefficient reflecting the incremental instability (potential for blocking) of assigning agent $i$ to resource $j$.
• Auxiliary objective functions $d_{ij}$ reflect secondary metrics such as travel distance, and terms involving $|\mathcal{P}_{<}(i, j)|$ count how close the assignment is to the top of agent $i$'s preference list.

The optimization then aims to

$$\min_{x_{ij} \in \{0,1\}} \lambda_1 \sum_{i,j} b^*_{ij} x_{ij} + \lambda_2 \sum_{i,j} d_{ij} x_{ij} + \lambda_3 \sum_{i,j} \frac{|\mathcal{P}_{<}(i,j)|}{|\mathcal{P}(i)|} x_{ij}$$

subject to feasibility and capacity constraints, and potentially an explicit constraint on the number of permitted blocking pairs: $\sum_{i,j} b^*_{ij} x_{ij} \leq B$ where $B$ is a prespecified upper bound on instability acceptable in the application setting. Similar terms can capture other objectives (e.g., matching couples in residency).

The vector $\lambda = (\lambda_1, \lambda_2, \lambda_3, ...)$ tunes trade-offs between stability and alternate goals. Practically, policy designers explore this space via grid search to obtain Pareto-efficient solutions (Bertsimas et al., 17 Sep 2025).

3. Inverse Optimization to Encode Stability

A central technical challenge lies in representing instability as a tractable objective (since the set of all blocking pairs is enormous in large-scale problems). The algorithm resolves this via inverse optimization:

• Run DA to produce a stable reference solution $x^{DA}$.
• Recover a cost vector $b^*$ such that $x^{DA}$ is optimal for an assignment LP with costs $b^*$. This is accomplished by solving

$$\min_{b}~ b^\top x^{DA} + \lambda \|b - \bar{b}\|_2$$

subject to dual feasibility, complementary slackness, and strong duality equations.

• The resulting $b^*$ implicitly penalizes deviations from stability (assignments whose inclusion would convert $x^{DA}$ into an unstable match).

This approach avoids directly listing all blocking pair constraints, enabling scalability to thousands of agents and programs.

4. Empirical Performance and Trade-offs in Practice

Computational experiments demonstrate clear trade-offs:

• In synthetic Boston Public Schools data, the optimization-based method reduced average student travel distance by up to 20% with only a 5% increase in blocking pairs compared to standard DA. Increasing the number of students matched beyond the DA maximum incurred negligible additional instability.
• In a synthetic residency setting with couples (approx. 12% of applicants), allowing 5–10% blocking pairs increased the proportion of couples placed at the same location by up to 10% over DA.
• Distributions of matched students' preference ranks remained similar to DA, suggesting that relaxing stability need not harm individual preference realization if appropriately weighted in the objective.

These findings establish that a controlled relaxation of stability can yield substantial improvements in operational metrics that are policy-relevant.

5. Mathematical Summary and Model Formulation

The key mathematical formulation comprises two coupled problems:

• Inverse optimization for stability penalties:

  minimize  b^T x^{DA} + λ‖b – b̄‖₂ subject to  u_i + v_j + w ≤ b_{ij} ∀ (i,j) (b_{ij} – u_i + v_j – w) x_{ij}^{DA} = 0 ∀ (i,j) ∑_{i,j} b_{ij} x_{ij}^{DA} = ∑_i u_i + ∑_j C_j v_j + Nw ∑_{i,j} b_{ij} = 1

• Multi-objective assignment with stability costs:

  minimize  λ₁ ∑_{i,j} b_{ij}^* x_{ij} – λ₂ ∑_{i,j} [|𝒫_{<}(i,j)|/|𝒫(i)|] x_{ij} – λ₃ ∑_{i,j} [|𝒬_{<}(i,j)|/|𝒬(j)|] x_{ij} subject to ∑_{j∈𝒫(i)} x_{ij} ≤ 1 ∀ i ∑_{i∈𝒬(j)} x_{ij} ≤ C_j ∀ j ∑_{i,j} x_{ij} ≥ N ∑_{i,j} d_{ij} x_{ij} ≤ T x_{ij} ∈ {0,1} ∀ (i,j)

Analogous formulations are provided for other settings, such as enforcing a lower bound on couples matched to the same hospital: $$\sum_{i\in\mathcal{R}} \sum_{j \in \mathcal{P}(i)} x_{ij} x_{D(i)j} \geq T$$ where $D(i)$ denotes the partner of applicant $i$.

6. Practical Consequences and Limitations

The optimization-based relaxation of stability provides new flexibility to designers of assignment mechanisms:

• Policy flexibility: Administrators can explicitly choose the acceptable trade-off between system-wide objectives and stability, rather than being forced to accept DA's rigid trade-off.
• Computational scalability: By using inverse optimization to encode blocking pair penalties and avoiding explicit enumeration, the methodology scales to large assignment markets.
• Domain adaptation: The approach applies to diverse contexts—including school assignment, medical residency, and potentially any matching market with many-to-one structure and side constraints—so long as preferences, capacities, and alternate metrics are quantifiable.

Limiting assumptions include the requirement for explicit preference lists, known capacities, and acceptability of some controlled instability. The method does not address settings with fully endogenous preferences or strategic behavior beyond the assignment phase, and the practical meaning of “acceptable” instability will be context-dependent.

7. Summary and Future Prospects

Optimization-based many-to-one assignment algorithms articulated in this framework relax full stability in a controlled and quantifiable manner, optimizing simultaneously for alternative performance criteria such as cost minimization, increased match rates, and applicant satisfaction (Bertsimas et al., 17 Sep 2025). By combining inverse optimization (to encode stability) with tractable multi-objective assignment solvers, these algorithms make it possible to efficiently explore the Pareto frontier between instability and secondary objectives in large-scale markets—a significant advance over classical DA-based assignment exclusively focused on stability. Potential research directions include quantifying long-run effects of stability relaxations in repeated settings, exploring endogenous or strategic choice models, and extending the approach to more complex matching-with-contracts or multi-period assignment environments.