Equitability Theorem: Matroid Partitioning, SODA 2026

• The Equitability Theorem is a result on k-base partitionable matroids that guarantees each base’s intersection with any subset differs by at most one.
• It employs iterative pairwise exchanges using a directed exchange graph and multi-layered potential functions to achieve equilibrium in polynomial time.
• The theorem underpins applications in fair division, offering EF1 and MMS guarantees under diverse valuation settings in practical allocation problems.

Equitability Theorem (SODA 2026)

The Equitability Theorem, as settled in SODA 2026 by Akrami–Raj–Végh, establishes that for any matroid whose ground set can be partitioned into $k$ bases, and for any subset $S$ of the ground set, there exists a partition into $k$ bases such that the cardinalities of their intersections with $S$ differ by at most one. This result, which proves a long-standing conjecture of Fekete and Szabó, also extends to multiple disjoint subsets and has notable applications in matroid-constrained fair division, including instance-optimal guarantees for envy-free up to one item (EF1) and maximin share (MMS) allocations in settings with additive (bi- or tri-valued) valuations (Akrami et al., 16 Jul 2025).

1. Formal Statement and Key Definitions

Let $M = (E, \mathcal{I})$ be a matroid of rank $r$ on finite ground set $E$. Denote its set of bases by $\mathcal{B}$, and write $\mathrm{rk}(X)$ for the rank function. A matroid is $k$-base–partitionable if $E$ can be partitioned into $k$ disjoint bases.

Equitability Theorem (Theorem 1.1):

If $E$ can be partitioned into $k \geq 1$ disjoint bases and $S \subseteq E$ is arbitrary, then there exists a partition $E = B_1 \cup \cdots \cup B_k$ with $B_i \in \mathcal{B}$ for all $i$ such that

$$\left\lfloor\frac{|S|}{k}\right\rfloor \leq |B_i \cap S| \leq \left\lceil\frac{|S|}{k}\right\rceil,$$

equivalently,

$$\max_{i,j} \left|\,|B_i \cap S| - |B_j \cap S|\,\right| \leq 1.$$

Such a partition can be constructed in polynomial time given an independence oracle.

A class of matroids is $\ell$-equitable if, for any $k$-base partition and $\ell$ arbitrary disjoint subsets $S_1, \dots, S_\ell \subseteq E$, a $k$-base partition can be found so that each $S_i$ is split as evenly as possible—equitable if $\ell=1$.

2. Proof Structure and Exchange Operations

The proof proceeds in two main stages:

• Exchange between two bases: Given disjoint bases $B_1, B_2$ with $|B_1 \cap S| < |B_2 \cap S|$, construct a small exchange set $X \subseteq B_1 \cup B_2$ to increment $|B_1 \cap S|$ by 1 and decrement $|B_2 \cap S|$ by 1, maintaining the base property. This involves defining a directed exchange graph $D(B_1, B_2)$ encoding single-element exchanges preserving basality. A key lemma ensures the existence of a chordless cycle (the $(t,S)$-exchangeable set) in this graph, whose symmetric difference effects the desired rebalancing.
• Iterated pairwise balancing: Starting from any $k$-base partition, repeated pairwise rebalancing between the bases with the highest and lowest $|B_i \cap S|$ reduces the max-min gap, halting when the difference becomes at most 1. Since each round can be implemented in polynomial time and at most $|S|$ steps are needed, the process is efficient.

3. Two-set Extension and Optimality

An extension addresses the case of two disjoint sets $S_1, S_2$. For $b^1_i = |B_i \cap S_1|$ and $b^2_i = |B_i \cap S_2|$, there exists a $k$-base partition such that simultaneously: $$\max_{i,j}|b^1_i - b^1_j| \leq 1, \qquad \max_{i,j}(|b^1_i - b^1_j| + |b^2_i - b^2_j|) \leq 2, \qquad \max_{i,j}|(b^1_i + b^2_i) - (b^1_j + b^2_j)| \leq \max_{i,j}|b^1_i - b^1_j|.$$ For $k=2$, this yields the tight bounds $|B_1 \cap S_1| - |B_2 \cap S_1| \leq 1$ and $|B_1 \cap S_2| - |B_2 \cap S_2| \leq 2$. The proof uses a multi-layered potential function to ensure that each exchange reduces overall disparity lexicographically.

A counterexample in the $K_4$ graphic matroid demonstrates that these bounds are tight—one cannot hope for $|b^1_i - b^1_j| \leq 1$ for both sets in arbitrary matroids.

4. Polynomial-time Rebalancing Algorithms

A constructive, efficient rebalancing algorithm is provided:

Algorithm overview:

1. While the gap $\max_i |B_i \cap S| - \min_j |B_j \cap S| \geq 2$, identify the indices $i$ (min), $j$ (max).
2. Perform the two-base exchange to find a set $X$ causing $|(B_i \Delta X) \cap S| = |B_i \cap S| + 1$.
3. Replace $B_i, B_j$ with $B_i \Delta X, B_j \Delta X$.
4. Repeat until the disparity is at most 1.

Implementation relies on constructing the directed exchange graph for the pair $(B_i, B_j)$, searching for an appropriate exchange cycle using BFS, and applying the exchange.

Complexity is polynomial in $|E|$ (number of items) and the number of independence oracle calls.

5. Applications to Matroid-Constrained Fair Division

The theorem enables new results for fair division under matroid constraints, in scenarios where feasible allocations correspond to $|N|$-base partitions.

• Envy-freeness up to one item (EF1) for identical tri-valued valuations: If all agents share the valuation $v:E\to\{0, a, b\}$, define $S_H = \{g: v(g) = b\}$, $S_L = \{g: v(g) = a\}$. The two-set extension ensures that each agent's share of high-value and low-value items differ by at most 1 and 2, respectively. Basic case analysis verifies EF1. For $b \geq 2a$, the partition itself is EF1; when $b < 2a$, further exchanges on zero-valued items resolve residual envy, ensuring existence in polynomial time (Akrami et al., 16 Jul 2025).
• Maximin share (MMS) guarantee for bi-valued additive valuations: With $v_i(g) \in \{0, a_i\}$, partitioning $E$ into $n$ bases guarantees each agent $i$ receives at least $\mu_i^n(E) = v_i(E)/n$. The construction uses the lone-divider protocol and Hall's theorem, finding a bipartite matching or assigning blocking sets inductively. This method ensures an MMS allocation under matroid constraints for all agents, computable in polynomial time (Akrami et al., 16 Jul 2025).

6. Significance and Broader Impact

The SODA 2026 Equitability Theorem fully resolves the conjecture that any $k$-base partitionable matroid admits, for any subset $S$, a $k$-base partition equitably splitting $S$. The polynomial-time constructiveness supports scalable algorithms with applications to scheduling, allocation under combinatorial constraints, and resource division. The two-set extension delineates the best-possible multi-objective equitability achievable for arbitrary matroids.

Theoretically, these results establish robust links between combinatorial optimization in matroids and equitable resource division. Practically, they underwrite fair division protocols for real-world constraints—EF1 under tri-valued valuations, and exact MMS for bi-valued settings—demonstrating the power of matroidal equitability as a unifying fairness concept.

7. Connections and Further Directions

Related exchange properties—including the multiple exchange and Equitability-Exchange—are generalized in algebraic frameworks relying on Grassmann–Plücker identities for representable matroids. These algebraic tools yield polynomial-time algorithms for equitable partitions, facilitate generalizations to more complex valuation structures, and guide ongoing research in combinatorial allocation and optimization (Oki et al., 20 Nov 2025).

The Equitability Theorem forms a cornerstone for a broad range of algorithmic, fair division, and algebraic investigations over matroidal domains, both in foundational theory and in computational practice.