Bipartite matching under communication constraints

Abstract: In modern data center networks, thousands of hosts contend for shared link capacity; the scale of these systems makes centralized scheduling impractical. This article models such scheduling as a bipartite matching problem under communication constraints: senders express interest in forming connections, and receivers respond using only locally available information. A class of single-round probabilistic matching algorithms is proposed, built on two key ideas: degree-biased sampling, in which senders use receiver degrees to inform their random selection, and random thinning, in which senders report only a random subset of their connections. Analytical performance guarantees are established for random graph models. In sparse regimes, degree-biased sampling yields a higher expected matching size than prior communication-constrained algorithms; in denser settings, a counterintuitive phenomenon emerges where deliberately restricting available connections through thinning increases the expected number of matches. Combining thinning to degree two with greedy selection produces an algorithm that requires no parameter tuning and, in packet-level simulations with production traffic traces, significantly extends the network stability region. Although motivated by data center network scheduling, the underlying framework of bipartite matching under local information constraints is portable to other resource allocation settings.

• The paper presents decentralized matching algorithms leveraging degree-biased sampling and random thinning to maximize sender–receiver pairings under a one-shot communication model.
• It rigorously analyzes both uniform and greedy selection methods with closed-form derivations and simulations validated on realistic data center workloads.
• The study demonstrates that limiting candidate choices with the 2CGS approach can improve network stability and increase throughput by up to 20% compared to existing protocols.

Detailed Analysis of "Bipartite matching under communication constraints" (2604.10744)

Problem Formulation and Motivation

This paper investigates decentralized bipartite matching under stringent communication constraints, with specific deployment motivation in modern data center networks (DCNs) where thousands of hosts require scalable, high-throughput communication scheduling. The fundamental challenge addressed is efficient sender–receiver pairing in scenarios where centralized scheduling is infeasible and each node possesses only local, partial graph information. The communication constraint is formalized by restricting message exchanges to at most one round, with senders relying solely on local two-hop views and limited feedback from receivers. The edges of the bipartite graph represent feasible sender–receiver transmissions for "long messages" (bulk data), and the objective is to maximize the matching—the number of communication pairs—using only minimal, distributed coordination.

Algorithmic Contributions

The core contribution resides in the design and analysis of a class of single-round probabilistic decentralized matching algorithms:

• Degree-Biased Sampling (DB($\alpha$)): At the heart is a family of randomized algorithms parameterized by $\alpha \leq 0$, where each sender stochastically selects a receiver neighbor, biasing its choice according to the degrees of the recipients (favoring low-degree receivers for $\alpha < 0$). This mechanism leverages the "power-of-choices" effect shown to mitigate contention but tailored for the one-shot bipartite matching context.
• Random Thinning: Novel is the explicit use of random subsampling at the NOTIFY stage, wherein each sender limits its candidate neighbor set by a thinning operation (either binomial or capped cardinality), inducing a "sparsified intention graph". Counterintuitively, in denser network regimes, reducing the candidate degree available to senders increases the expected matching size due to lower collision rates at receivers, a phenomenon the paper formally characterizes.
• 2-Choice Greedy Selection (2CGS): They empirically identify "thinning to degree two (+ greedy selection for receivers of lowest degree)" as a universally robust instantiation—realistic for practical DCN workloads due to its non-parameterized, tuning-free operation.

Theoretical Results

A suite of rigorous performance analyses is developed:

• Uniform Selection Baseline ($\alpha=0$): The mean matching fraction is derived in closed form for i.i.d. degree distributions of the feasible (intention) graph. It is proven that for DB(0), the expected matching depends only on the probability of a node holding at least one edge, not on higher degree moments—a strong robustness property.
• Greedy Selection ($\alpha=-\infty$): For the extreme case of always favoring minimum-degree receivers, a recursive distributional analysis yields a non-trivial, computable lower bound for expected matching size in the sparse (Poissonian) regime. Theoretical predictions are verified by extensive simulations, with bounds shown to be tight for practically sized graphs ($N \sim 100$).
• Asymptotic Independence Justification: The paper rigorously justifies the use of independence assumptions among sender events in the large system limit, via concentration of degree distributions, simplifying analysis but with quantifiable error terms.

Empirical Evaluation

The analytic findings are substantiated by simulation studies on random bipartite graphs with both binomial and deterministic (constant) out-degree distributions, examining matching size as a function of mean sender degree and algorithmic parameter $\alpha$. Key findings include:

• Thinning Benefits: For both graph ensembles, the empirical matching size peaks at intermediate (not maximal) mean degree, and over-thickening (higher sender degree) is shown to degrade decentralized matching performance except in the centralized case. The optimal performance is consistently achieved by capping sender degrees to two (max(2)-thinning).
• Parameter Robustness: With practical DCN workloads, the 2CGS variant delivers matching fractions (normalized) of .68–.74, notably outperforming uniform and naive greedy selection (and exceeding existing one-round protocols such as dcPIM and iSLIP in throughput and stability margin).

Packet-Level System Simulations

To validate practical efficacy, the algorithms are embedded in detailed packet-level simulations (YAPS) using both real and synthetic data center workloads:

• Key performance metrics: normalized throughput, stability region width, and flow completion time (FCT) for both short and long messages.
• The 2CGS algorithm significantly extends the network stability region (maximum sustainable load before unbounded latency queue buildup) by approximately 20% compared to dcPIM and iSLIP baselines.
• Control message overhead for 2CGS remains near parity with these baselines in stable regime, with order-of-magnitude reductions available at the onset of instability.

Theoretical and Practical Implications

The results have strong implications for both distributed algorithm theory and systems:

• Theoretically, the paper establishes that local, single-shot randomization—when combined with modest degree-awareness and judicious restriction of choice—can asymptotically approach the performance of multi-round and even centralized algorithms for large, sparse systems, challenging prior assumptions about the necessity of extensive coordination for effective resource allocation.
• Practically, these algorithms can be directly integrated into DCN stacks and other resource allocation systems where communication is both costly and time-bounded (e.g., high-frequency trading, cloud job allocation), being parameter-agnostic and robust to the highly heterogeneous degree distributions endemic to modern infrastructure.
• The counterintuitive result that less information (or artificially limited choice) can yield more scalable, collision-averse matching in dense regimes is likely to inform further work on distributed matching and load balancing.

Future Directions

Potential avenues for extension include:

• Adaptive or learning-based algorithms that can dynamically tune thinning and selection bias in response to observed workload heterogeneity.
• Generalization to multi-round protocols with formal tradeoffs among rounds, message complexity, and matching optimality.
• Applying degree-biased decentralized algorithms in non-networked application domains (e.g., large-scale market matching, distributed storage migration).

Conclusion

This paper rigorously demonstrates that, under severe communication constraints, careful randomization and degree biasing yield high-performance distributed matching algorithms that are broadly applicable. The work closes several gaps in the literature regarding single-round distributed matching, especially under structural (rather than preference-based) constraints typical in high-throughput networked systems.

Strong claims are:

• The expected matching fraction under uniform selection is degree distribution-insensitive.
• Thinning to degree two combined with local greedy selection is universally robust and parameter-free.
• Empirically, such procedures enlarge the stability region by up to 20% in DCN-scale simulations compared to best prior one-round approaches.

These findings will serve as the foundation for robust, communication-efficient distributed scheduling protocols in large-scale networked systems.