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：摘要：设计同时支持高吞吐量和低延迟的路由方案是网络优化的核心挑战之一。这类路由可形式化为$h$长度流，其定义为流路径长度不超过$h$的流。许多被广泛研究的算法基元（如最大及最大长度约束不相交路径）是$h$长度流的特例。同样，最优$h$长度流是网络优化中的基本量，它刻画了网络完成多种分布式基元的速度（精确到多对数因子）。本文首次给出了计算$(1 - \epsilon)$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约束不相交路径问题给出了高效的确定性CONGEST算法——解决了Chang和Saranurak（FOCS 2020）提出的开放问题——同时给出了最大长度约束不相交路径的本质上最优的并行和分布式近似算法。前者极大简化了用于计算扩展图分解的确定性CONGEST算法。我们还利用我们的技术，首次在CONGEST中给出了二分图$b$匹配的高效$(1-\epsilon)$近似算法。最后，基于流算法，我们首次实现了高效计算$h$长度割匹配，这是近期长度约束扩展图分解进展的核心对象。