[论文解读] An Efficient Algorithm for All-Pairs Bounded Edge Connectivity

Shyan Akmal, Ce Jin|arXiv (Cornell University)|Jan 1, 2023

Complexity and Algorithms in Graphs被引用 1

一句话总结

该论文提出了首个在有向图中求解 k-有界所有点对边连通性（k-APC）问题的算法，时间复杂度为 Õ((kn)^ω)，显著快于对 k ≥ 3 时求解完整的所有点对连通性问题。关键洞见在于：在代数框架中，用低秩矩阵求逆替代完整矩阵求逆，从而实现对所有连通性小于 k 的点对高效计算有界边连通性。

ABSTRACT

Our work concerns algorithms for a variant of Maximum Flow in unweighted graphs. In the All-Pairs Connectivity (APC) problem, we are given a graph G on n vertices and m edges, and are tasked with computing the maximum number of edge-disjoint paths from s to t (equivalently, the size of a minimum (s,t)-cut) in G, for all pairs of vertices (s,t). Over undirected graphs, it is known that APC can be solved in essentially optimal n^{2+o(1)} time. In contrast, the true time complexity of APC over directed graphs remains open: this problem can be solved in Õ(m^ω) time, where ω ∈ [2, 2.373) is the exponent of matrix multiplication, but no matching conditional lower bound is known. Following [Abboud et al., ICALP 2019], we study a bounded version of APC called the k-Bounded All Pairs Connectivity (k-APC) problem. In this variant of APC, we are given an integer k in addition to the graph G, and are now tasked with reporting the size of a minimum (s,t)-cut only for pairs (s,t) of vertices with min-cut value less than k (if the minimum (s,t)-cut has size at least k, we can just report it is "large" instead of computing the exact value). Our main result is an Õ((kn)^ω) time algorithm solving k-APC in directed graphs. This is the first algorithm which solves k-APC faster than simply solving the more general APC problem exactly, for all k ≥ 3. This runtime is Õ(n^ω) for all k ≤ poly(log n), which essentially matches the optimal runtime for the k = 1 case of k-APC, under popular conjectures from fine-grained complexity. Previously, this runtime was only achieved for general directed graphs when k ≤ 2 [Georgiadis et al., ICALP 2017]. Our result employs the same algebraic framework used in previous work, introduced by [Cheung, Lau, and Leung, FOCS 2011]. A direct implementation of this framework involves inverting a large random matrix. Our new algorithm is based off the insight that for solving k-APC, it suffices to invert a low-rank random matrix instead of a generic random matrix. We also obtain a new algorithm for a variant of k-APC, the k-Bounded All-Pairs Vertex Connectivity (k-APVC) problem, where for every pair of vertices (s,t), we are now tasked with reporting the maximum number of internally vertex-disjoint (rather than edge-disjoint) paths from s to t if this number is less than k, and otherwise reporting that this number is at least k. Our second result is an Õ(k²n^ω) time algorithm solving k-APVC in directed graphs. Previous work showed how to solve an easier version of the k-APVC problem (where answers only need to be returned for pairs of vertices (s,t) which are not edges in the graph) in Õ((kn)^ω) time [Abboud et al, ICALP 2019]. In comparison, our algorithm solves the full k-APVC problem, and is faster if ω > 2.

研究动机与目标

• 开发一种更快的算法，用于计算有向图中的有界边连通性，尤其针对 k ≥ 3 的情形，此前的方法速度并不快于完整 APC 问题。
• 弥合 k = 1,2 时已知最优时间复杂度与 k ≥ 3 时当前最先进方法在 k-APC 问题上的差距。
• 通过证明低秩矩阵求逆足以支持 k-APC，扩展 Cheung、Lau 和 Leung（FOCS 2011）提出的代数框架。
• 为 k-有界所有点对点连通性（k-APVC）提供一种新算法，其时间复杂度优于以往工作。

提出的方法

• 基于矩阵求逆的代数框架来编码连通性信息，该框架最初由 Cheung、Lau 和 Leung（FOCS 2011）提出。
• 将 m × m 随机矩阵的求逆替换为 (k+1) × (k+1) 低秩矩阵的求逆，大幅降低计算成本。
• 采用流向量框架通过矩阵的秩来计算连通性值，其中导出矩阵 F_{s,t} 的秩决定了每对点 (s,t) 的 min(k, λ(s,t))。
• 通过将矩阵 D_{ij} 构造为五个 n × n 矩阵的乘积，利用快速矩阵乘法高效计算 F_{s,t} 的元素。
• 对所有点对应用并集界，以确保基于秩的连通性估计具有高概率正确性。
• 通过为点不相交路径定义类似矩阵 B 和 C，将方法扩展至点连通性，从而得到时间复杂度为 Õ(k²n^ω) 的 k-APVC 算法。

实验结果

研究问题

• RQ1 对于有向图中 k ≥ 3 的情形，k-APC 是否可以比一般的所有点对连通性问题求解得更快？
• RQ2 是否可以实现 Õ(k²n^ω) 的时间复杂度求解 k-APC，与新提出的 k-APVC 算法保持一致？
• RQ3 能否将 APC 的已知 Õ(m^ω) 时间复杂度改进为 Õ((kn)^ω)，尤其当 k ≤ poly(log n) 时？
• RQ4 是否存在条件性下界，排除 k-APC 和 k-APVC 的更快算法？或者它们是否可能在 Õ(n^ω) 时间内求解，尤其当 k 适中偏大时？
• RQ5 能否为 k-APC 和 k-APVC 构建与已知 APVC 验证器时间复杂度相似的非确定性验证器？

主要发现

• 该论文提出了一种时间复杂度为 Õ((kn)^ω) 的 k-APC 算法，这是首次在有向图中对 k ≥ 3 实现该时间复杂度的算法。
• 当所有 k ≤ poly(log n) 时，该算法实现 Õ(n^ω) 时间复杂度，与在细粒度复杂度猜想下 k = 1 和 k = 2 的最优时间复杂度一致。
• 关键技术贡献在于证明 k-APC 仅需低秩矩阵求逆，从而将瓶颈从 m × m 矩阵求逆降低至 (k+1) × (k+1) 矩阵求逆。
• 对于 k-APVC，论文给出了时间复杂度为 Õ(k²n^ω) 的算法，当 ω > 2 时，其速度优于以往工作，并且解决了完整问题（不仅限于非边点对）。
• 两个算法的正确性均以高概率（至少 1 − 5/n）得到证明，通过在所有点对上应用并集界。
• 结果在广泛接受的细粒度复杂度猜想下，建立了新的、最优或近似最优的时间复杂度上界。