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.

翻译：设计既能支持高吞吐量又能实现低延迟的路由方案是网络优化的核心挑战之一。此类路径可形式化为 $h$-长度流，即流路径长度被限制为至多 $h$ 的流。许多经典算法原语——如最大和最大长度受限的不相交路径——都是 $h$-长度流的特例。同样，最优 $h$-长度流是网络优化中的一个基本量，它（在对数因子范围内）表征了网络完成多种分布式原语的速度。在本工作中，我们首次给出了计算 $(1 - \epsilon)$-近似 $h$-长度流的高效算法。我们提出了确定性算法，其并行时间为 $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$，分布式 CONGEST 时间为 $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}) \cdot 2^{O(\sqrt{\log n})})$。我们还给出了一种 CONGEST 算法，该算法以高概率成功且仅需 $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$ 时间。利用我们的 $h$-长度流算法，我们首次为最大长度受限不相交路径问题提供了高效的确定性 CONGEST 算法——解决了 Chang 和 Saranurak（FOCS 2020）提出的一个开放问题——以及针对最大长度受限不相交路径的近似最优并行和分布式近似算法。前者极大地简化了用于计算展开分解的确定性 CONGEST 算法。我们还使用我们的技术，首次在 CONGEST 中为二部图 $b$-匹配问题提供了高效的 $(1-\epsilon)$-近似算法。最后，利用我们的流算法，我们首次提出了高效计算 $h$-长度割匹配的算法，这是长度受限展开分解近期进展中的核心对象。