Distributed Load Balancing on Unrelated Machines

Abstract: We study the well-known load balancing problem in the distributed CONGEST model of computation. We consider the unrelated machines setting, where each job $j$ specifies a size $s_{ij}$ for every machine $i$. We want to find an assignment $\varphi: J \to M$ minimizing the maximum machine load, where the load of a machine $i$ is the total size of the jobs assigned to it. In the CONGEST model, the state-of-the-art is an algorithm that runs in polylog rounds and returns a $(1+\varepsilon)$-approximate fractional solution from Ahmadian, Liu, Peng, and Zadimoghaddam (2021). However, this algorithm, as well as all previous CONGEST algorithms only solve a special case of load balancing, where each job has the same size on each machine. Our main contribution is an algorithm for general sizes $s_{ij}$. The algorithm computes a $(1+\varepsilon)$-approximate fractional solution or a $(2+\varepsilon)$-approximate integral solution in polylog rounds. The problem structure changes significantly once we allow arbitrary edge-sizes, so our techniques are very different from those used in previous algorithms for distributed load balancing. One ingredient of our result is a black-box tool of independent interest: a $(1+\varepsilon)$-approximation algorithm to arbitrary mixed packing-covering linear programs in the CONGEST model in polylog rounds. such algorithms were known in the more powerful parallel model, but previous polylog-round algorithms in the distributed CONGEST model only solved pure packing or pure covering problems. We improve upon a recent $O(D\,\mathrm{polylog})$-round CONGEST algorithm for mixed packing-covering, where $D$ is the diameter of the communication graph.

• The paper introduces a novel distributed MixPC solver that achieves a (1+ε)-approximation for mixed packing-covering LPs in polylogarithmic rounds.
• The authors leverage the MixPC subroutine to design a deterministic (2+ε)-approximation algorithm for load balancing on unrelated machines, matching centralized lower bounds.
• The approach employs geometric search, flow freezing, and local bucketing techniques to overcome global coordination barriers in the CONGEST model.

Distributed Load Balancing on Unrelated Machines in the CONGEST Model

Problem Formulation and Motivation

This paper addresses classical makespan minimization (load balancing) on unrelated machines in the distributed CONGEST model. The instance graph $G=(J \cup M,E)$ is bipartite: nodes correspond to jobs $J$ and machines $M$, and for each admissible assignment, there is an edge $(i,j)$ with a nonnegative size $s_{ij}$. The objective is to compute an assignment $\varphi: J \to M$ that minimizes the maximum load $\max_i \sum_{j: \varphi(j) = i} s_{ij}$. This is the most general form of the load balancing problem, with no restrictions on the assignment matrix $[s_{ij}]$ beyond non-negativity.

The distributed CONGEST model restricts communication in each synchronous round to $O(\log n)$ bits per edge in the communication (input) graph. Historically, distributed approximation schemes only handled restricted assignments ($s_{ij} \in \{1,\infty\}$) or cases where all of a job’s admissible assignments have equal size ($J$0), see e.g., Ahmadian et al., 2021. However, the general unrelated machines case—where $J$1 values are fully arbitrary—introduces substantial complexity, both from the perspective of combinatorial structure and distributed computation.

Main Contributions

1. Distributed $J$2-Approximate Mixed Packing-Covering (MixPC) LPs

The paper develops a polylogarithmic-round distributed algorithm in CONGEST for mixed packing-covering linear programs with arbitrary diameter and topology. Prior distributed CONGEST solutions either (a) only handled pure packing or covering (not MixPC) [Bartal et al., 2004], or (b) for MixPC, exhibited at least $J$3 round complexity (where $J$4 is the communication diameter) for feasibility (Vos et al., 30 Apr 2026). The presented algorithm simultaneously enforces all mixed packing and covering constraints to within a $J$5 factor, in $J$6 rounds, without requiring global broadcasts. This uses an adaptation of Young’s parallel MixPC scheme [cs/0205039], replacing explicit computation of global softmin/softmax with a locally synchronized threshold value incremented at fixed rates. This technique eliminates the need for $J$7-round graph-wide computations—previously a core obstacle for distributed MixPC feasibility algorithms.

2. Distributed $J$8-Approximate Load Balancing for Unrelated Machines

By leveraging the distributed MixPC approximation as a module, the authors provide a deterministic, polylogarithmic-round CONGEST algorithm for integral unrelated machine load balancing that achieves a $J$9-approximation. Notably, this matches (up to $M$0-slack) the known polynomial-time centralized hardness threshold of 2, as shown by Lenstra-Shmoys-Tardos [Lenstra et al., 1990], and is tight under an unconditional distributed lower bound ($M$1 for approximation ratio below 2) (Burke et al., 2018). The algorithm computes a fractional $M$2-approximation first, then applies a distributed rounding scheme (inspired by [Li 2023, (Li, 2021)]) to produce an explicit job assignment.

The authors’ approach introduces an interpolation-based "flow freezing" paradigm to deal with the inherent non-locality of infeasibility in CONGEST, which prevents direct application of classic centralized "guess-and-check" thresholding protocols. By locally freezing certain sets of jobs and machines (within radius $M$3 of infeasibility witnesses) and interpolating between successive fractional solutions, the algorithm sidesteps the global coordination barriers and achieves geometric search and convergence on the optimal value.

Technical Approach

MixPC Algorithm

The MixPC solver is a distributed adaptation of Young’s smooth-potential method for mixed packing and covering LPs [cs/0205039]. The main challenges lie in the infeasibility of global computation of potential values (softmax/softmin), due to bandwidth and locality restrictions of CONGEST. To overcome this, nodes advance a local potential threshold $M$4, exponentially growing at a fixed rate. This threshold is guaranteed (by monotonicity and the properties of the update rule) to never exceed the global potential, ensuring feasibility without any explicit graph-wide reduction. Inner rounds focus on local eligibility checks, using only incident data, thus enabling fully distributed operation with strong round complexity.

Unrelated Machine Load Balancing via Geometric Search, Flow Interpolation, and Bucketing

The general unrelated case fundamentally breaks standard distributed expansion arguments and augmenting-path truncations that suffice for restricted assignments. The authors develop a bucketing and interval-shifting framework to decompose the edge-size matrix locally (for each job) into intervals of bounded aspect ratio. This allows each job to ignore excessively heavy edges without significant loss in the overall makespan. For each bucket, the relevant subinstance can be solved with bounded ratio, so solutions are combined by averaging over randomly shifted bucketing decompositions. Interpolation-based flow freezing ensures that feasibility persists at boundaries, even as local failures arise in the geometric search process. Rounding then uses distributed matching and the expansion properties of the virtual assignment graph to guarantee integral assignments remain within $M$5-approximation.

Numerical/Complexity Guarantees and Implications

• Fractional solution: $M$6-approximation in $M$7 CONGEST rounds.
• Integral solution: $M$8-approximation in the same round complexity.
• The algorithm is independent of the graph diameter and does not require global knowledge of edge-size range ($M$9), achieved through bucketing and randomized interval decomposition.
• The lower bound ($(i,j)$0 for $(i,j)$1-approximation) rules out sub-2 approximations with less than linear-in-diameter complexity; this algorithm matches that barrier.

Theoretical and Practical Impact

The presented framework closes a significant gap in distributed scheduling—achieving the optimal centralized approximation guarantee for unrelated machines in the restrictive CONGEST model with only $(i,j)$2 round overhead. The MixPC subroutine is of independent use; any CONGEST-encodable LP that can be cast as mixed packing-covering can now be solved up to slack $(i,j)$3 in a small number of rounds regardless of the communication diameter.

Practically, this enables deployment of globally near-optimal job assignments in massive systems with severe bandwidth and locality limitations, across a host of networked scheduling, resource allocation, and matching tasks. The techniques (flow interpolation, local bucketing and intervalization, and distributed potential maintenance) offer a blueprint for generalized distributed optimization for other combinatorial or LP-based problems over arbitrary topologies.

Prospective Directions

Potential extensions and impact include:

• Adaptation of the black-box MixPC solver to non-bipartite or more general assignment and resource allocation structures.
• Tight lower bounds and complexity tradeoffs for broader classes of distributed LPs.
• Empirical assessment of round complexity and communication requirements in real-world, large-scale deployments.
• Integration with streaming, online, and dynamic distributed systems, leveraging the robust locality and approximation properties of the presented framework.

Conclusion

This work establishes the first polylogarithmic-round distributed approximation scheme for general unrelated-machine load balancing in the CONGEST model, matching both centralized approximability and distributed lower bounds. The hierarchical "bucketing" and local flow-averaging ideas, together with the distributed MixPC machinery, break the prior dependence on global coordination and uniformity. The results substantially enlarge the class of optimization problems on massive distributed graphs that are efficiently and robustly solvable in practice and theory, enabling fine-grained scheduling and assignment under minimal global communication.