Student-Friendly Matching Module

• The paper presents a novel matching module that balances Pareto efficiency and fairness by leveraging an acquaintance graph to enforce local envy-freeness.
• The module formalizes a many-to-one matching environment integrating student preferences, school capacities, and social relationships to enable localized fairness checks.
• It offers tractable algorithms for tree and bounded treewidth graphs, providing clear trade-offs between efficiency and controlled levels of local envy.

A student-friendly matching module, as formalized and implemented in “A New Relaxation of Fairness in Two-Sided Matching Respecting Acquaintance Relationships” (Takeshima et al., 21 Aug 2025), is a mechanism for assigning students to schools (or other resources) that optimally balances efficiency and fairness, with a particular focus on mitigating envy among students who are socially connected or acquainted. The module is built on a standard many-to-one matching formalism augmented by an acquaintance graph, and introduces the concept of local envy-freeness, which relaxes global fairness constraints by localizing attention to envy occurring only between acquaintances. This approach yields tractable and implementable algorithms in several structurally meaningful cases and provides a parameterized trade-off between efficiency (Pareto optimality) and fairness (envy-freeness level).

1. Formal Model: Matching with Acquaintance Graphs

The matching environment consists of a set of students $S = \{s_1, ..., s_n\}$ and a set of schools $C = \{c_1, ..., c_m\}$, each with a capacity $q(c) \in \mathbb{N}$. Matchings $\mu \subseteq S \times C$ associate each student to at most one school, respecting all school capacities.

Each student $s$ provides a strict preference order $\succ_s$ over $C \cup \{\emptyset\}$ (including the possibility of remaining unmatched), while each school $c$ provides a strict priority order $\succ_c$ over $S \cup \{\emptyset\}$.

Crucially, an undirected acquaintance graph $G = (S, E)$ encodes social relationships: an edge $\{s, s'\} \in E$ indicates $s$ and $s'$ are acquaintances, with $N(s)$ denoting the set of all $s'$ adjacent to $s$.

2. Fairness Metrics: Envy-Freeness and Localizations

Justified envy—the classic fairness violation—occurs when student $s$ wishes to exchange matches with acquainted $s'$ at school $c$ (i.e., $c \succ_s \mu(s)$ and $s \succ_c s'$), with $s$ not assigned to $c$ but $s'$ is. The global notion (envy-freeness) often conflicts with efficiency.

Local envy restricts consideration to pairs $(s, s') \in E$, so only acquaintances’ assignments can generate actionable envy. The local envy set is

$$\text{LocalEnvy}(\mu) = \left\{(s,s') \in E : \mu(s') = c \neq \mu(s),\ c \succ_s \mu(s),\ s \succ_c s' \right\}.$$

This framework is generalized by introducing $k$-local envy-freeness (LEF$_k$): for each $s$, at most $k$ elements of $N(s)$ can induce local envy under $\mu$.

3. Pareto Efficiency and Compatibilities

A matching $\mu$ is Pareto-efficient (PE) if there is no other feasible $\mu'$ making every student at least as happy and some strictly happier. Full envy-freeness and PE are often incompatible, motivating the relaxation to local constraints.

Importantly, in unrestricted graphs, the decision problem of whether a PE+LEF$_0$ matching exists is NP-complete, and thus generally computationally intractable. However, meaningful graph restrictions open efficient algorithmic avenues.

4. Algorithmic Solutions and Special Cases

4.1 Trees with Single-Peaked Preferences: B-LT2

If $G$ is a tree and all school priorities are single-peaked (the set of top $t$ students is always connected), the B-LT2 mechanism achieves PE + LEF$_0$:

• Step 1: Match all mutually-best pairs.
• Step 2: Iteratively assign each unassigned $s$ to her favorite available school $c$ if $c$ prefers $s$ to all unassigned neighbors.
• Step 3: For residuals, pair and assign students attacking each other at their top choices.
• Termination: $O(n^2 m)$ steps; outputs PE and LEF$_0$ by design.

4.2 Bounded Treewidth Graphs

For $G$ of treewidth $k$, the B-LT($k+1$) variant achieves PE + LEF$_{k-1}$, modifying eligibility in Step 2 to allow $s$ to be among top $k$ unassigned neighbors. Under general preferences, serial dictatorship with a $k$-degenerate master list yields PE + LEF$_k$.

Complexity: Sorting and ordering $O(n m)$, assignment steps $O(n \log m)$.

5. Trade-Offs, Parameter Choices, and Guidelines

• $k=0$ yields maximal local fairness at the potential expense of efficiency.
• Increasing $k$ progressively relaxes local fairness, enhancing efficiency but tolerating limited, localized envy.
• In sparse or tree-like graphs (e.g., school districts by geography/social circles), small $k$ (1–3) suffices for strong fairness.
• Construction recommendations:
  • Build $G$ using neighborhood or peer data.
  • Set $k$ near half the average degree of $G$.
  • Begin with $k=0$ for small instances; relax $k$ for larger ones as needed.

6. Illustrative Example

Consider $S = \{A,B,C,D,E\}$, $C = \{1,2,3,4\}$, $q(1..4) = 1$, and $G$ as a path $A$–$B$–$C$–$D$–$E$. Students and schools have given single-peaked priorities.

Running B-LT2, only $B$ and $D$ match in Step 2; Step 3 pairs $C$ and $A$ via mutual attacks, leaving $E$ unmatched. This matching is PE, and LEF$_0$ holds for the path structure.

7. Practical Implementation and Data Structures

• Graph Representation: Adjacency list for $G$ (per student).
• Preferences: Student and school priorities as ordered lists or arrays enabling $O(1)$ comparisons.
• Capacities: Integer counters per school.
• UI Integration: Modular pipeline for input parsing, graph construction, matching execution, and output formatting. Web-based deployment can leverage class data or surveys to build $G$, present envy diagnostics, and allow real-time $k$ tuning.

Simulation tools should permit configuration and reporting of residual envy counts to inform trade-offs.

Summary Table: Tradeoff Regions for PE + LEF$_k$ Algorithms

Graph Class	School Priority	Achievable Fairness
Tree	Single-peaked	PE + LEF$_0$
Treewidth-$k$	Single-peaked	PE + LEF$_{k-1}$
Treewidth-$k$	General	PE + LEF$_k$
Unrestricted	Any	NP-complete (no guarantee)

Conclusion

The student-friendly matching module built on local envy-freeness leverages graph-based social acquaintance information to relax classic global fairness constraints. Special-case algorithms deliver PE + LEF$_k$ in polynomial time for tree and low-treewidth graph structures, with practical parameterizations enabling transparent calibration of the fairness-efficiency trade-off in real-world school-choice or similar contexts. The provision for bounding local envy along the acquaintance graph permits the implementation of matchings that are both socially palatable for students and algorithmically scalable.