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Group Fairness and Multi-criteria Optimization in School Assignment

Santhini K. A., Kamesh Munagala, Meghana Nasre, Govind S. Sankar

TL;DR

The paper addresses fair allocation of $n$ students to $m$ schools with capacities under group fairness constraints across $g$ demographic groups. It introduces a convex-programming relaxation combined with two rounding schemes to maximize a concave fairness function of group utilities while controlling capacity violations, and extends the method to arbitrary covering constraints and ranking optimization. The main results include polynomial-time and $n^{O(g)}$-time algorithms with capacity violations bounded by $O(g)$ and $O(g^2)$ respectively, as well as generalizations to multi-constraint settings and monotone constraints, yielding exact or near-exact fairness guarantees such as proportionality and Nash welfare. The framework is complemented by NP-hardness results and an empirical ILP benchmark showing practical performance far better than the worst-case bounds, along with a simulation study for ranking and weak dominance of ranks. Overall, the work provides a scalable, flexible approach to group-fair allocation in assignment problems with broad applicability to multi-criteria optimization and ranking tasks.

Abstract

We consider the problem of assigning students to schools, when students have different utilities for schools and schools have capacity. There are additional group fairness considerations over students that can be captured either by concave objectives, or additional constraints on the groups. We present approximation algorithms for this problem via convex program rounding that achieve various trade-offs between utility violation, capacity violation, and running time. We also show that our techniques easily extend to the setting where there are arbitrary covering constraints on the feasible assignment, capturing multi-criteria and ranking optimization.

Group Fairness and Multi-criteria Optimization in School Assignment

TL;DR

The paper addresses fair allocation of students to schools with capacities under group fairness constraints across demographic groups. It introduces a convex-programming relaxation combined with two rounding schemes to maximize a concave fairness function of group utilities while controlling capacity violations, and extends the method to arbitrary covering constraints and ranking optimization. The main results include polynomial-time and -time algorithms with capacity violations bounded by and respectively, as well as generalizations to multi-constraint settings and monotone constraints, yielding exact or near-exact fairness guarantees such as proportionality and Nash welfare. The framework is complemented by NP-hardness results and an empirical ILP benchmark showing practical performance far better than the worst-case bounds, along with a simulation study for ranking and weak dominance of ranks. Overall, the work provides a scalable, flexible approach to group-fair allocation in assignment problems with broad applicability to multi-criteria optimization and ranking tasks.

Abstract

We consider the problem of assigning students to schools, when students have different utilities for schools and schools have capacity. There are additional group fairness considerations over students that can be captured either by concave objectives, or additional constraints on the groups. We present approximation algorithms for this problem via convex program rounding that achieve various trade-offs between utility violation, capacity violation, and running time. We also show that our techniques easily extend to the setting where there are arbitrary covering constraints on the feasible assignment, capturing multi-criteria and ranking optimization.
Paper Structure (18 sections, 14 theorems, 21 equations, 1 table, 4 algorithms)

This paper contains 18 sections, 14 theorems, 21 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model
  3. Our Results
  4. Techniques and Related Work.
  5. Approximation Algorithm: Proof of \ref{['thm:main']}
  6. Convex Program Relaxation
  7. Generalized Assignment Rounding
  8. Improved Capacity Violation via Cake Frosting
  9. Generalization to Arbitrary Constraints
  10. Generalizing \ref{['thm:main1']}
  11. Generalizing \ref{['thm:main2']}
  12. Better Bounds for Monotonic Constraints
  13. Application: Weak Dominance of Ranks.
  14. ILP Benchmark and Simulation Study
  15. Empirical Results for School Assignment
  16. ...and 3 more sections

Key Result

Theorem 1

Given any monotone, concave fairness function $f$, let $\vec{U^*}$ be the utilities in the optimal solution. Then, there exist algorithms to compute an assignment $\vec{y}$ that satisfies relaxed school capacities $\vec{C'}$ and yields utilities $\vec{U'}$ with $U'_k \ge U^*_k$ for all groups $k \in

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof : Proof of \ref{['thm:main1']}.
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 12 more