Multilevel Fair Allocation in Matroid-Rank Domains

• Multilevel fair allocation is a framework that uses matroid and weighted rank functions to model layered fairness in resource distribution.
• It leverages combinatorial algorithms, including matroid intersection and greedy swaps, to optimize multiple fairness criteria like EF1, MMS, and Nash welfare.
• Practical applications span online scheduling, network coding, and fair assignations in education and markets, demonstrating its versatility and efficiency.

Multilevel fair allocation refers to frameworks and algorithms for assigning resources among agents in which the agents’ valuations are rank functions of matroids or, more generally, their weighted analogues. These structures underlie a vast array of allocation models, including discrete fair division, resource scheduling, network coding, and online assignment, and form one of the most tractable and expressive subfamilies in the broader theory of submodular set functions. Multilevel refers to the layered nature of fairness constraints—allocations may be optimized simultaneously for several criteria (e.g., utilitarian social welfare, Nash welfare, maximin share guarantees, EF1, pairwise fairness), often necessitating combinatorially rich and technically nuanced algorithms. This article presents a rigorous treatment of the foundational theory, computational methods, online models, and advanced fairness concepts in multilevel fair allocation over matroid-rank domains.

1. Matroid Rank Functions: Structure and Characterization

A matroid is a pair $M=(E, \mathcal{I})$, with $E$ a finite ground set and $\mathcal{I} \subseteq 2^E$ a nonempty family of sets, closed under subset and satisfying the exchange axiom: for any $X,Y \in \mathcal{I}, |X|<|Y|$, there exists $e\in Y\setminus X$ with $X\cup\{e\}\in\mathcal{I}$ (Barman et al., 2021). The rank function $r:2^E\to\mathbb{N}$ associates to each $S\subseteq E$ the size of the largest independent subset contained in $S$: $$r(S) = \max\{\,|I|\,:\, I\subseteq S, I\in\mathcal{I}\,\}.$$ Matroid rank functions are normalized ($r(\emptyset)=0$), integer-valued, monotone ($r(S)\le r(T)$ if $S\subseteq T$), and submodular ($r(A)+r(B)\ge r(A\cup B)+r(A\cap B)$ for all $A,B\subseteq E$). They satisfy the binary marginals property: for any $S\subseteq E$ and $j\notin S$, $r(S\cup\{j\})-r(S)\in\{0,1\}$ (Viswanathan et al., 2022).

Weighted Matroid Rank Functions generalize this to accommodate non-uniform element values: given $w: E\rightarrow \mathbb{R}_{\ge 0}$, the weighted function is

$$r_{M,w}(S) = \max\left\{\,\sum_{e\in I} w_e\,:\, I\subseteq S, I\in\mathcal{I}\right\}$$

with the same monotonicity and submodularity (Buchbinder et al., 2012). These functions always lie on extreme faces of the submodularity cone (Rashid et al., 2022).

2. Fairness Notions and Multilevel Guarantees

Table: Fairness Criteria under Matroid-Rank Valuations

Criterion	Guarantee under Matroid-Rank	Algorithmic Feasibility
EF1	Always achievable (Benabbou et al., 2020, Barman et al., 2021)	Polytime via matroid intersection + swaps (Benabbou et al., 2020)
MMS	Exact allocation exists (Barman et al., 2020)	Polytime: matroid-union + local exchanges (Barman et al., 2020)
PMMS	Exact allocation exists (Barman et al., 2020)	Polytime via augmentation (Barman et al., 2020)
Leximin/Nash	Coincides with EF1 (Benabbou et al., 2020)	Polytime via matroid flow or balanced allocations (Benabbou et al., 2020)
EFX	Partial positive results (Benabbou et al., 2020)	Exists in restricted cases

Meaning of criteria:

• EF1: Envy-freeness up to one good—no agent envies another after removal of a single good from the other's allocation.
• MMS: Maximin share—each agent receives at least the best she could guarantee for herself by partitioning the goods and receiving the least valued part.
• PMMS: Pairwise maximin share—stronger, considering joint shares for each agent pair.

Allocations maximizing utilitarian social welfare can satisfy stringent fairness simultaneously. The Nash welfare-maximizing and leximin allocations are both EF1 for matroid-rank functions (Benabbou et al., 2020), and the General Yankee Swap framework delivers optimality for a wide class of fairness functions (Viswanathan et al., 2022).

3. Algorithms and Exchange Properties

Multilevel fair allocation algorithms hinge on combinatorial properties of matroids and their strong exchange theorems:

• Matroid Union and Local Exchanges: Exact MMS and PMMS allocations are obtained by constructing the union matroid of agents' matroids, finding a maximum cardinality independent set (utilitarian welfare maximization), followed by local augmentations along shortest exchange paths to restore fairness (Barman et al., 2020, Viswanathan et al., 2022).
• Augmenting Path/Swap Frameworks: Because marginal gains are binary, allocation improvements can be reduced to exchanging goods along paths in an exchange graph, guaranteeing polynomial convergence (Viswanathan et al., 2022).
• Matroid-Intersection and Flow Techniques: For special subclasses (e.g., graphical or transversal matroids), optimal allocations can be extracted from network flow or matroid intersection algorithms in strongly polynomial time (Benabbou et al., 2020).
• Greedy Algorithms and H-Matroids: The greedy method is optimal in every allowable subset if and only if the rank function satisfies normalization, unit increase, and an H-extension property, characterizing H-matroids (Sano, 2011).

Key augmentation lemmas guarantee the feasibility and progress in iterative fairness-improving algorithms (Viswanathan et al., 2022).

4. Online and Polyhedral Models

In online and fractional settings, matroid-rank structure enables provably competitive algorithms:

• Online Weighted Rank Maximization: For online sequences of (possibly weighted) matroid-rank functions, randomized algorithms based on multiplicative updates and randomized rounding achieve competitive ratios within polylogarithmic factors of the offline optimum, exploiting the uncrossing of tight sets in the matroid polytope (Buchbinder et al., 2012).
• Online Submodular Assignment: When agent utilities are matroid ranks, online submodular welfare maximization admits a $(1-1/e)$-competitive integral allocation via a RANKING-style random-priority algorithm, generalizing online matching frameworks (Hathcock et al., 2024).
• Correlation Gap: The correlation gap of a (weighted) matroid-rank function is minimized under uniform weights and admits strict lower bounds (parametrized by rank and girth), directly tightening the guarantees in submodular maximization, contention resolution, and mechanism design (Husić et al., 2022).

The matroid polytope’s structural properties—such as principal partitions—govern the algorithmic tractability of online and convex-analytic resource allocation (Hathcock et al., 2024).

5. Advanced Theoretical Extensions

The matroid-rank paradigm admits several powerful extensions:

• Entropic and Information-Theoretic Connections: Binary matroid rank functions arise exactly as integer-valued mutual information functions for linear deterministic multiple-access channels; more broadly, weighted rank functions of matroids are entropic for integer weights over binary-representable matroids (Abbe, 2010, Rashid et al., 2022).
• Supermatroids and Modular Lattices: The axioms for matroid rank extend to supermatroids on modular lattices, with the rank function characterized by normalization, unit increment along covering pairs, and directional DR-submodularity. Classical matroid submodularity appears as a special case (Maehara et al., 2020).
• Powerful Sets: The rank function of a powerful set (every element is a power of $2$) coincides with a binary matroid rank if and only if it is subcardinal. This generalizes the discrete rank function paradigm to broader combinatorial contexts (Jones, 2020).

These connections allow the embedding of combinatorial optimization, information theory, and convex geometry within the multilevel fair allocation framework.

6. Structural Limits and Separations

The multilevel fairness property is robust for matroid-rank functions, but collapses for immediate supersets:

• Beyond Matroid-Rank: For binary XOS functions and weighted rank valuations, exact MMS allocations may not exist—matroid-rank is maximal among submodular function classes for which strong fairness and efficiency can be guaranteed in polynomial time (Barman et al., 2020, Benabbou et al., 2020).
• Strategyproofness and Mechanism Design: For matroid-rank valuations, mechanisms can achieve envy-freeness up to one good (EF1) with group-strategyproofness, but no Pareto-efficient, index-oblivious, truthful mechanism can guarantee maximin shares simultaneously (Barman et al., 2021).
• Correlation Gap: The $1-1/e$ bound tight for simple matroid-rank functions is strictly improved for matroids with girth greater than $2$ (Husić et al., 2022).

Thus, matroid-rank valuations establish a tractable frontier for multilevel fair allocation—admitting strong algorithms, precise fairness, and rich generalizations, while exposing sharp limits at their boundary.

7. Applications and Impact

Multilevel fair allocation over matroid-rank utility domains models a wide array of problems:

• Indivisible goods division with capacity or coverage constraints
• Online job or resource assignments with combinatorial caps
• Sensor/scheduling/network design subject to diverse independence structures
• Online advertising and welfare markets with submodular/coverage utility functions
• Fair course, residency, or matching assignments in institutional settings

Efficient algorithmic frameworks and strong theoretical guarantees now exist for utilitarian, Nash, leximin, EF1, MMS, and other fairness objectives on matroid-rank–valued domains, leveraging exchange-theoretic combinatorics, polyhedral algorithms, and online convex optimization (Benabbou et al., 2020, Barman et al., 2020, Buchbinder et al., 2012, Viswanathan et al., 2022, Hathcock et al., 2024). These results illuminate the unique appropriateness of the matroid-rank class for multilayered and dynamic fairness in algorithmic allocation.