Maximum Length-Constrained Flows and Disjoint Paths: Distributed, Deterministic and Fast

TL;DR

This paper introduces the first efficient deterministic and randomized algorithms for computing approximate $h$-length flows, disjoint paths, and related network optimization primitives in distributed and parallel settings, advancing the state-of-the-art in network routing and decomposition.

Contribution

It provides the first efficient algorithms for approximate $h$-length flows and solves open problems in length-constrained disjoint paths within distributed models.

Findings

Deterministic algorithms run in $ ilde{O}( ext{poly}(h, 1/\epsilon))$ parallel time.

Distributed algorithms succeed with high probability in $ ilde{O}( ext{poly}(h, 1/\epsilon))$ time.

Applications include improved algorithms for disjoint paths, bipartite $b$-matching, and length-constrained expander decompositions.

Abstract

Computing routing schemes that support both high throughput and low latency is one of the core challenges of network optimization. Such routes can be formalized as $h$-length flows which are defined as flows whose flow paths are restricted to have length at most $h$. Many well-studied algorithmic primitives -- such as maximal and maximum length-constrained disjoint paths -- are special cases of $h$-length flows. Likewise the optimal $h$-length flow is a fundamental quantity in network optimization, characterizing, up to poly-log factors, how quickly a network can accomplish numerous distributed primitives. In this work, we give the first efficient algorithms for computing $(1 - \epsilon)$-approximate $h$-length flows. We give deterministic algorithms that take $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$ parallel time and $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}) \cdot 2^{O(\sqrt{\log n})})$ distributed CONGEST time. We also give a CONGEST algorithm that succeeds with high probability and only takes $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$ time. Using our $h$-length flow algorithms, we give the first efficient deterministic CONGEST algorithms for the maximal length-constrained disjoint paths problem -- settling an open question of Chang and Saranurak (FOCS 2020) -- as well as essentially-optimal parallel and distributed approximation algorithms for maximum length-constrained disjoint paths. The former greatly simplifies deterministic CONGEST algorithms for computing expander decompositions. We also use our techniques to give the first efficient $(1-\epsilon)$-approximation algorithms for bipartite $b$-matching in CONGEST. Lastly, using our flow algorithms, we give the first algorithms to efficiently compute $h$-length cutmatches, an object at the heart of recent advances in length-constrained expander decompositions.