---
title: 'Equitability Theorem: Matroid Partitioning, SODA 2026'
url: https://www.emergentmind.com/topics/equitability-theorem-soda-2026
type: topic
---

# Equitability Theorem: Matroid Partitioning, SODA 2026

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 [2507.12100].

## 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 [2507.12100].

- **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 [2507.12100].

## 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 [2511.16021].

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.

Source: https://www.emergentmind.com/topics/equitability-theorem-soda-2026