Matroid-Rank Utility Functions

• Matroid-rank utility functions are defined by the matroid's rank, capturing independence and hierarchical constraints in resource allocation.
• They exhibit key properties such as normalization, monotonicity, submodularity, and binary marginals that facilitate efficient greedy and exchange-based optimizations.
• These functions underpin fairness guarantees in multilevel allocation, enabling computation of EF1, MMS, and Nash welfare allocations using advanced algorithmic paradigms.

Multilevel Fair Allocation refers to a family of discrete resource allocation problems in which indivisible items (goods) are allocated among agents, subject to hierarchical (multilevel) feasibility and fairness constraints. The multilevel aspect typically arises from combinatorial feasibility structures—often modeled by matroids, partitioned resources, or layered eligibility restrictions—while the fair allocation component seeks to reconcile objectives such as maximin share (MMS), envy-freeness up to one good (EF1), Nash social welfare (NSW), and other justice criteria, often under strategic agent behaviors and in the absence of monetary transfers.

1. Hierarchical Feasibility and Matroid-Rank Utilities

At the core of multilevel fair allocation is the matroid—a combinatorial abstraction of independence with hierarchical structure. Given a ground set $M$ of indivisible goods, a matroid $\mathcal M = (M, \mathcal I)$ prescribes feasible allocations via its independent sets $\mathcal I$, characterized by hereditary and augmentation properties. The associated rank function $r: 2^M \rightarrow \mathbb N$ is defined by

$$r(S) = \max\{ |I| : I \subseteq S, I \in \mathcal I \}$$

and serves as a canonical utility function for agents facing feasibility constraints that capture quotas, partitioned categories, forestry problems, and matching-type eligibility (Barman et al., 2021, Buchbinder et al., 2012, Viswanathan et al., 2022). The "multilevel" nature can manifest as layered matroids—partition, graphic, transversal—yielding a hierarchy of allocation restrictions.

Weighted extensions $$f_w(S) = \max_{I \subseteq S,\, I \in \mathcal I} \sum_{e \in I} w_e$$ admit further granularity, modeling settings where item values vary per agent or per layer (Buchbinder et al., 2012, Rashid et al., 2022).

2. Core Properties and Structural Axioms

Matroid-rank utility functions exhibit key structural properties fundamental to their role in multilevel fair allocation:

• Normalization: $r(\emptyset) = 0$
• Monotonicity: $S \subseteq T \Rightarrow r(S) \le r(T)$
• Submodularity (Diminishing Returns): $r(A) + r(B) \ge r(A \cup B) + r(A \cap B)$
• Binary Marginals: $r(S \cup \{j\}) - r(S) \in \{0,1\}$

These properties enable efficient exchange arguments, augmenting-path techniques, and greedy optimization. In the multilevel context, extensions to modular and distributive lattice settings (supermatroids, H-matroids (Sano, 2011, Maehara et al., 2020)) preserve these properties via local extension or directional submodularity axioms, facilitating greedy-optimality in hierarchical feasibility spaces.

Weighted variants $r_{M,w}$ remain submodular and located on extreme faces of the submodularity cone (Rashid et al., 2022). Entropic representability further connects these functions to entropy cones under additional representability or weight conditions.

3. Fairness Criteria and Algorithmic Guarantees

Multilevel fair allocation studies several prominent fairness notions, which, for matroid-rank and related utilities, often align with tractable algorithmic and structural properties:

• Envy-Freeness up to One Good (EF1):
  • For matroid-rank valuations, EF1 allocations exist, are socially optimal, and can be computed efficiently via matroid-intersection routines followed by envy-induced transfer subroutines (Benabbou et al., 2020, Barman et al., 2021).
  • Mechanisms such as Prioritized Egalitarian and leximin/Nash welfare maximizers provide group strategy-proofness and Pareto efficiency (Barman et al., 2021, Viswanathan et al., 2022).
• Maximin Share (MMS) and Pairwise Maximin Share (PMMS):
  • Existence and computation of MMS and PMMS allocations are always guaranteed for matroid-rank utilities via matroid-union algorithms and local exchange (Barman et al., 2020, Viswanathan et al., 2022).
  • Immediate extensions to weighted ranks or binary XOS valuations can destroy these guarantees, indicating a sharp boundary of tractability (Barman et al., 2020).
• Nash Social Welfare (NSW) and Leximin:
  • Leximin and Nash welfare objectives can be attained in polynomial time; for certain matroidal subclasses, specialized network-flow formulations apply (Benabbou et al., 2020, Viswanathan et al., 2022).
• Mechanism Design Without Money:
  • Truthful multilevel fair allocation mechanisms align with the property of gradualness: monotonicity of output bundle sizes under single-good removals, underlying the possibility of robust allocation rules in the absence of payments (Barman et al., 2021).

4. Algorithmic Paradigms for Multilevel Fair Division

Algorithmic advances in multilevel fair allocation leverage two main ingredients:

• Matroid Union and Exchange Graphs: Union matroid constructions enable simultaneous welfare optimization subject to multiple (possibly agent-specific) matroid independence constraints; exchange graphs encode transfer possibilities along efficient augmenting paths for correcting fairness violations (Barman et al., 2020, Viswanathan et al., 2022).
• Yankee Swap and Path-Augmentation: Binary marginal structure of matroid-rank functions allows path-augmentation techniques (General Yankee Swap frameworks), in which each fairness-improving step is discrete (unit-sized), enabling efficient cycling through agents to optimize various justice criteria (Viswanathan et al., 2022).
• Fractional Relaxation and Randomized Rounding: Fractional LP relaxations based on matroid polytopes, with subsequent randomized rounding using uncrossing-closure properties of tight sets, underlie competitive online and offline algorithms for maximizing rank-based objectives under matroid constraints (Buchbinder et al., 2012).

A summary of algorithmic paradigms and their structural requirements appears in the following table:

Paradigm	Structural Assumption	Key Feature
Matroid Union	Matroid ranks per agent	Exact MMS/PMMS/social welfare; polytime
Envy-induced Transfer	Binary marginals	USW-optimal EF1; strong fairness-exchange
Yankee Swap	Binary submodular	Leverages unit-marginals for fairness steps
Fractional + Rounding	Weighted rank / polytope	LP relaxations, randomized rounding

5. Multilevel Structure in Online and Mechanism Design

Online variants of multilevel fair allocation address the arrival of items or objectives over time. Here, matroid-based LP relaxations and hierarchical randomized rounding yield competitive guarantees (Buchbinder et al., 2012, Hathcock et al., 2024):

• Online Welfare Maximization: Matroidal RANKING algorithms achieve a $(1-1/e)$-competitive ratio for online submodular assignment with matroid-rank utilities, leveraging principal partitions and the water-filling paradigm generalized to matroidal settings (Hathcock et al., 2024).
• Correlation Gap Analysis: The matroidal structure allows improved bounds for the correlation gap beyond the generic $1-1/e$, concretely parameterized by rank and girth, with implications for posted pricing and contention resolution (Husić et al., 2022).
• Mechanism Design Under Strategic Agents: Group strategy-proofness is achievable for EF1 but not for MMS in matroid-rank settings, indicating that multilevel constraints fundamentally affect the interplay between truthfulness, fairness, and efficiency (Barman et al., 2021).

6. Extensions, Boundaries, and Open Directions

The multilevel fair allocation framework—grounded in matroid (and weighted matroid) rank functions—strikes a maximal balance between structural richness and algorithmic tractability. Immediate generalizations (binary XOS, weighted ranks) admit more expressive models but lose key guarantees, such as existence of exact MMS allocations (Barman et al., 2020). Extensions to supermatroids on modular lattices (Maehara et al., 2020) and H-matroids (Sano, 2011) capture broader hierarchically-structured feasibility regimes, yet retain optimization characterizations tied to the shape of rank increments and submodularity.

A plausible implication is that further progress on multilevel fair allocation will require characterizing those intermediate valuation classes and feasibility structures that preserve algorithmic tractability of key fairness notions. Open problems persist in quantitative tightness of competitive ratios under composite or non-matroidal constraints, robust mechanism design for broader valuation classes, and the full integration of entropy-theoretic methods into utility design (Rashid et al., 2022, Abbe, 2010).

7. Connections to Information Theory and Optimization

Recent work connects matroid-based multilevel allocation to extremal dependence structures in information theory:

• Entropy as Utility: Integral weighted matroid-rank functions represent entropy functions under suitable representability conditions, motivating the use of entropy regions as a geometric tool for analyzing allocation feasibility (Rashid et al., 2022).
• Mutual Information and Matroid Ranks: The rank function of a (binary) matroid corresponds precisely to the uniform mutual information function of a multiple access channel, revealing deep structural analogues between combinatorial allocation and network information flow (Abbe, 2010).

This confluence suggests that multilevel fair allocation with matroid-rank utilities not only unifies a range of allocation and optimization models but also aligns with the geometric and algebraic structure of extremal dependencies in broader mathematical domains.