一种用于最小费用流的次平方双方通信协议
A Subquadratic Two-Party Communication Protocol for Minimum Cost Flow
AI总结：
本文将顺序内点法适配到双方通信模型，给出双侧约束线性规划算法，并据此得到最大流与最小费用流的 $\ ilde{O}(n^{1.5})$ 比特通信协议。
摘要：
本文讨论双方通信模型中的最大流问题：两方各自持有一个公共顶点集上的边子集，目标是以最少的通信计算并图的最大流。我们证明该问题可以用 $\ ilde{O}(n^{1.5})$ 比特通信解决，改进了平凡的 $\ ilde{O}(n^2)$ 上界。为实现这一点，我们推导出两个更一般的结果：1. 我们提出一种用于双侧约束线性规划的随机算法；当每个约束至多有 $k$ 个非零项时，该算法需要 $\ ilde{O}(n^{1.5}k)$ 比特通信。该结果改进了 [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24] 的先前工作；他们针对单侧约束 LP 达到 $\ ilde{O}(n^2)$ 比特复杂度。更精确分析后，其算法对单侧约束 LP 可达到 $\ ilde{O}(n^{1.5}+nk)$ 的比特复杂度。尽管如此，对于稀疏矩阵，我们的方法与这一复杂度匹配，同时将适用范围扩展到双侧约束。2. 利用该结果，我们证明最小费用流问题作为求解双侧约束线性规划的特例以及最大流问题的推广，也能够以 $\ ilde{O}(n^{1.5})$ 比特的通信复杂度求解。这些结果通过将 [van den Brand, Lee, Liu, Saranurak, Sidford, Song, Wang, STOC'21] 在顺序设置中用于求解双侧约束 LP 的基于内点法（IPM）的算法适配到双方通信模型而获得。该适配使用了 [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24] 为分布式凸优化开发的技术。
英文摘要：
In this paper, we discuss the maximum flow problem in the two-party communication model, where two parties, each holding a subset of edges on a common vertex set, aim to compute the maximum flow of the union graph with minimal communication. We show that this can be solved with $\tilde{O}(n^{1.5})$ bits of communication, improving upon the trivial $\tilde{O}(n^2)$ bound. To achieve this, we derive two additional, more general results: 1. We present a randomized algorithm for linear programs with two-sided constraints that requires $\tilde{O}(n^{1.5}k)$ bits of communication when each constraint has at most $k$ non-zeros. This result improves upon the prior work by [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24], which achieves a complexity of $\tilde{O}(n^2)$ bits for LPs with one-sided constraints. Upon more precise analysis, their algorithm can reach a bit complexity of $\tilde{O}(n^{1.5} + nk)$ for one-sided constraint LPs. Nevertheless, for sparse matrices, our approach matches this complexity while extending the scope to two-sided constraints. 2. Leveraging this result, we demonstrate that the minimum cost flow problem, as a special case of solving linear programs with two-sided constraints and as a general case of maximum flow problem, can also be solved with a communication complexity of $\tilde{O}(n^{1.5})$ bits. These results are achieved by adapting an interior-point method (IPM)-based algorithm for solving LPs with two-sided constraints in the sequential setting by [van den Brand, Lee, Liu, Saranurak, Sidford, Song, Wang, STOC'21] to the two-party communication model. This adaptation utilizes techniques developed by [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24] for distributed convex optimization.
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