Max-Min Fairness in Resource Allocation

• Max-min fairness is a concept in resource allocation that maximizes the minimum utility of users through lexicographic optimality.
• It finds applications in wireless networks, MIMO communications, and combinatorial optimization, ensuring equitable rate distribution among entities.
• Algorithmic strategies such as convex relaxations, fixed-point iterations, and alternating optimization provide effective approximations despite NP-hard challenges in discrete settings.

Max-min fairness is a foundational concept in resource allocation, routing, networking, multiuser communication, and combinatorial optimization. Its central objective is to maximize the minimum utility, rate, or performance achieved across all entities (e.g., users, flows, providers, or nodes), with multiple variants spanning continuous and discrete domains, convex and nonconvex settings, and deterministic and stochastic optimization perspectives.

1. Formal Definition and Lexicographic Optimality

Max-min fairness refers to allocation rules that seek to maximize the utility of the worst-off participant, then—subject to that constraint—maximize the second-worst, and so on (lexicographic maximization). In canonical convex settings, an allocation $(x_1,\ldots,x_n)$ is max-min fair if no $x_i$ can be increased without strictly decreasing some $x_j\leq x_i$, or without violating feasibility.

In rate allocation problems, this is often formulated as: $\max_{x\in\mathcal{X}}\min_{i=1,\ldots,n}~ u_i(x)$ with $\mathcal{X}$ denoting the feasible set and $u_i$ the utility/rate function for user $i$—subject to constraints such as capacity, power budgets, or combinatorial allocations. The resulting optimal value is sometimes called the max-min value.

Many formulations allow for equivalent lexicographic characterizations: the set of max-min fair allocations is the set of lexicographically maximal feasible allocations.

2. Computational Complexity and Hardness

Max-min fairness problems exhibit a wide spectrum of computational complexity, critically depending on domain and constraint structure:

• Continuous Convex Settings: In classical flow-based resource allocation or convex wireless power control (with continuous variables and convex constraints), max-min fairness can often be solved efficiently via bisection, water-filling, or convex optimization.
• Mixed-Integer/Discrete Settings: In assignment, matching, routing, resource allocation with indivisible items, or joint base-station association and power control, the problem becomes mixed-integer and nonconvex, often exhibiting strong NP-hardness.

For example, the joint base-station association and power control problem under per-base-station power constraints maximized for worst-user SINR is proven NP-hard by reduction from 3-SAT, with the nonconvexity rooted in discrete association variables and interdependent power budgets (Sun et al., 2014). In MIMO interfering broadcast channels, the max-min rate maximization is also NP-hard with at least two antennas at each node (Razaviyayn et al., 2012).

Combinatorial problems such as restricted max-min fair allocation (Santa Claus problem) are fundamentally hard: one can estimate the optimum to within a factor $4+\epsilon$ via Configuration LP relaxations, but polynomial-time combinatorial approximation ratios were only reduced to 13 (Annamalai et al., 2014).

3. Algorithmic Strategies: Decomposition, Relaxation, and Alternating Optimization

Despite intrinsic hardness, a variety of algorithmic paradigms yield optimality in special cases or effective approximations in practice.

3.1 Convex Relaxations and Fixed-Point Algorithms

For continuous settings or when constraints can be relaxed, convex optimization and fixed-point iterations can efficiently approximate the optimal value:

• Sum-Power Relaxation: In the downlink BS association/power control problem, relaxing per-BS constraints to a single sum-power budget enables a duality with an uplink version, solved by fast fixed-point iteration (ULSum), then projected back via another fixed-point map (M-mapping) to satisfy original per-BS constraints (Sun et al., 2014).
• Geometric Programming and Log-Domain Transformations: For massive MIMO uplink max-min fairness, the original geometric program can be reformulated in the log-domain, resulting in a smooth convex objective that is amenable to accelerated projected gradient methods, drastically reducing complexity versus traditional GP solvers (Farooq et al., 2020).
• Successive Convex Approximation (SCA): Nonconvex problems with coupled variables (such as power allocation in wave-domain and RSMA systems) are tackled by linearizing the nonconvex parts, iteratively solving a convex surrogate at each step (Quran et al., 13 May 2025).

3.2 Combinatorial and Discrete Methods

• Configuration LP and Local Search: In discrete max-min fair allocation, the best known integrality gap for the Configuration LP is $4+\epsilon$, but polynomial-time combinatorial algorithms (using Haxell-style hypergraph matchings, greedy α-edges, and lazy updates) achieve a 13-approximation (Annamalai et al., 2014).
• Assignment Reformulation: In the heterogeneous cellular BS-user association with $K=N$ and (high) SINR thresholds, the association subproblem reduces to a maximum-weight assignment problem with log-gains, solvable in polynomial time by the Hungarian or Auction algorithm (Sun et al., 2014).

3.3 Alternating Optimization and Block-Coordinate Methods

• Block-Coordinate Descent: For multiuser MIMO max-min fairness, problem structure allows alternating updates of user filters (receiver/precoder), covariance matrices, and auxiliary variables, reducing a nonconvex joint program to sequences of convex or closed-form subproblems (e.g., in weighted MMSE, AWMSE, or MM frameworks) (Razaviyayn et al., 2012, Joudeh et al., 2015, Naghsh et al., 2019).
• Minorization-Maximization (MM): The nonconvex rate functions can be minorized by affine surrogates at each iteration, resulting in convex programs (e.g., SOCP) with monotonic convergence to stationary points (Naghsh et al., 2019).
• Fractional Programming and Extragradient Methods: In rate-splitting multiple access, block-wise convexity is exploited by fractional programming to reparameterize the joint rate lower-bounds, and a variational inequality (VI) is solved via extragradient steps for each block (beamforming and common-rate shares) (Luo et al., 6 Jul 2025).

3.4 Specialized Algorithms for Robust or Secure Max-Min Fairness

• Semidefinite Relaxations (SDR) and S-Procedure: Robust max-min fairness for SWIPT, secure beamforming, or cognitive radio incorporates uncertainties in CSI via S-Procedure to convert infinite nonconvex constraints into finite-dimensional LMIs, then efficiently solves the resulting SDRs (with the relaxation being tight, i.e., optimal transactions are rank-one) (Zhou et al., 2016, Ng et al., 2014).
• One-dimensional Search: When a key variable appears in a nonconvex but unimodal fashion (e.g., the power splitting ratio in SWIPT), the max-min fairness objective can be optimized globally by one-dimensional grid search over that variable, with the inner problem being convex for each fixed value (Zhou et al., 2016, Kim et al., 2022).

4. Extensions: Stochastic, Online, and Learning-Based Max-Min Fairness

Recent work extends max-min fairness beyond static optimization, incorporating uncertainty, learning, and real-time adaptation:

• Online Learning of Demands: When users' resource requirements are unknown and only revealed through noisy or stochastic feedback, mechanisms inspired by classical water-filling alternate exploration and MMF-based exploitation, achieving asymptotically efficient, fair, and strategy-proof allocations with provable learning rates depending on feedback model (deterministic, parametric, or nonparametric) (Kandasamy et al., 2020).
• Provider and Group Fairness in Recommendation and Ranking: Max-min objectives are integrated into dual decomposition-based online re-ranking (P-MMF), lifting the worst-off provider's exposure efficiently in a low-dimensional dual space, and achieving $O(\sqrt{T})$ regret with computational costs independent of corpus size (Xu et al., 2023). In ranking, distributional max-min fairness over randomized mechanisms under group-quotas is achieved via ellipsoid methods with weighted-ranking oracles (Garcia-Soriano et al., 2021).
• Stochastic Wireless Resources: Statistical CSI, stochastic channel gains, or random arrivals are handled via upper bounds, sample-average approximation, or by optimizing expected max-min rates over stochastic models, using combinations of GP, gradient ascent, and convex upper-bounding (Ginige et al., 20 Apr 2025).

5. Max-Min Fairness in Application Domains

5.1 Wireless Networks and Multiuser Communication

Max-min fairness is crucial for fairness-centric scheduling and resource allocation in cellular, massive MIMO, SWIPT, NOMA, RSMA, and cognitive radio networks:

• Joint Power and Beamforming: Alternating between power allocation (via convex optimization or fractional programming) and multi-layer phase-only or wave-based beamforming (using Riemannian or gradient-based methods) is standard in advanced metasurface and rate-splitting systems (Quran et al., 13 May 2025, Ginige et al., 20 Apr 2025, Luo et al., 6 Jul 2025).
• Security and Energy Harvesting: Robust beamforming under max-min objectives delivers the worst-case secrecy or harvested energy via tight SDR (often guaranteeing rank-one optimality), leveraging S-procedure-driven reductions to SDPs (Zhou et al., 2016, Ng et al., 2014).
• Multiuser MIMO/IBC/IC: Max-min rate maximization under user co-channel interference is addressed via block coordinate or MM methods (Razaviyayn et al., 2012, Naghsh et al., 2019), with convergence or stationarity guarantees under suitable surrogates.

5.2 Combinatorial Allocation and Routing

• Resource Allocation: Max-min fairness appears in allocation with indivisible resources (restricted allocation), where combinatorial rounding and local search approximate lex-max allocations (Annamalai et al., 2014).
• Energy Harvesting Sensor Networks: Max-min fair rate assignment and routing under energy causality, battery, and flow constraints are achieved by water-filling, max-flow, and FPTAS-based approximation for fractional routing (Marasevic et al., 2014).
• Parallel Channels: In multichannel communication, max-min fair resource allocation is tightly characterized by the Lovász theta function and Delsarte bounds of an induced sharing graph, with no-odd-cycle properties yielding closed-form solutions (0808.0987).

6. Theoretical Characterization and Performance Bounds

Several domains admit explicit bounds on the achievable max-min value:

• Graph-theoretic Functionals: In Gaussian parallel channels, the max-min fair performance is determined by the Lovász function or Delsarte bound of a channel-sharing graph, modulated by the $2$-norm of a feasible performance vector; presence or absence of odd cycles governs the tightness of the bound (0808.0987).
• Convex Relaxation Gaps: The Configuration LP gap quantifies the difference between estimable max-min value and best-approximation in allocation problems—the current gap being between $4+\epsilon$ (estimate) and $13$ (best-known polynomial-time approximation ratio) (Annamalai et al., 2014).
• Price of Robustness: Imposing robust (worst-case) fairness under imperfect CSI or adversarial uncertainty typically reduces the worst-off utility, quantifying the performance-secrecy, performance-harvested energy, or latency-reliability trade-off (Zhou et al., 2016, Ng et al., 2014).

7. Controversies, Limitations, and Alternative Notions

Critiques of classical max-min fairness highlight susceptibility to manipulations (sybil attacks, e.g., splitting flows), small-packet attacks in networks, and insensitivity to congestion contribution (Miaji et al., 2010). Alternatives such as "maxmin-charge" fairness or scheduling mechanisms like Just Queueing have been proposed to account for congestion burden, penalizing cheaters via user- or flow-level attributes, though without the universal mathematical structure of max-min fairness.

• (Sun et al., 2014): Joint Downlink Base Station Association and Power Control for Max-Min Fairness: Computation and Complexity
• (Annamalai et al., 2014): Combinatorial Algorithm for Restricted Max-Min Fair Allocation
• (Razaviyayn et al., 2012): Linear Transceiver Design for a MIMO Interfering Broadcast Channel Achieving Max-Min Fairness
• (Zhou et al., 2016): Robust Max-Min Fairness Energy Harvesting in Secure MISO Cognitive Radio With SWIPT
• (Joudeh et al., 2015): Achieving Max-Min Fairness for MU-MISO with Partial CSIT: A Multicast Assisted Transmission
• (Naghsh et al., 2019): Max-Min Fairness Design for MIMO Interference Channels: a Minorization-Maximization Approach
• (Farooq et al., 2020): A Low-Complexity Approach for Max-Min Fairness in Uplink Cell-Free Massive MIMO
• (Quran et al., 13 May 2025): Max-Min Fairness in Stacked Intelligent Metasurface-Aided Rate Splitting Networks
• (Luo et al., 6 Jul 2025): An Efficient Max-Min Fair Resource Optimization Algorithm for Rate-Splitting Multiple Access
• (Kim et al., 2022): SWIPT-enabled NOMA in Distributed Antenna System with Imperfect Channel State Information for Max-Sum-Rate and Max-Min Fairness
• (Xu et al., 2023): P-MMF: Provider Max-min Fairness Re-ranking in Recommender System
• (Garcia-Soriano et al., 2021): Maxmin-Fair Ranking: Individual Fairness under Group-Fairness Constraints
• (0808.0987): A new graph perspective on max-min fairness in Gaussian parallel channels
• (Ng et al., 2014): Max-min Fair Wireless Energy Transfer for Secure Multiuser Communication Systems
• (Kandasamy et al., 2020): Online Learning Demands in Max-min Fairness
• (Ginige et al., 20 Apr 2025): Max-Min Fairness for Stacked Intelligent Metasurface-Assisted Multi-User MISO Systems
• (Marasevic et al., 2014): Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis
• (Miaji et al., 2010): Breaking the Legend: Maxmin Fairness notion is no longer effective

This article reflects the state of the art on the max-min fairness problem across algorithmic, theoretical, and applicative dimensions, rooted entirely in research visible in the arXiv corpus.