Finding Most Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time

TL;DR

This work delivers the first randomized near-linear time algorithms for listing all minimum vertex cuts of polylogarithmic size that yield at least three components (k-shredders) and for finding the most shattering one. The authors extend local flow techniques to operate on a local scale and combine them with a pairwise connectivity oracle under vertex failures to efficiently verify shredders, breaking the longstanding quadratic bottleneck of CT99. The main results show that, for k = O(polylog(n)), all k-shredders can be listed in time O(m + k^5 n log^4 n) with probability 1 - n^{-97}, and a most shattering minimum vertex cut can be produced in the same bound. The approach integrates edge-sampling for balanced shredders with localized Shredders and verification via connectivity oracles for unbalanced shredders, offering potential impact on vertex-connectivity augmentation and dynamic maintenance, and suggesting directions for further exploration in related Pfaffian and brace-related problems.

Abstract

We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has stood for more than two decades. Our work also removes a bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain near-linear time algorithms, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of connectivity oracles subject to vertex failures to speed up a static graph algorithm.

Figures (8)

• Figure 1: Let $\Pi$ be the set of three openly-disjoint simple paths from $x$ to $y$. A call to $\tt{Shredders}(x, y)$ will identify $N(Q_1) = N(Q_2)$ as potential $3$-shredders.
• Figure 2: Here we have $\Pi = \{\pi_1, \pi_2\}$. The candidate $\{s_1, s_2\}$ is straddled by the edge $(u, v)$ because $\delta_{\pi_2}(u) < \delta_{\pi_2}(s_2)$ and $\delta_{\pi_1}(v) > \delta_{\pi_1}(s_1)$.
• Figure 3: Suppose the $k$-shredder $S = \{s_1, s_2, s_3\}$ is captured by $(x, \nu, \Pi)$. We can recover the component $Q$ by exploring bridges of $\Pi$ using BFS.
• Figure 4: The existence of an $x {\rightsquigarrow} z$ path implies the existence of vertices $u$ and $v$ (it is possible that $x = u$ and $v = z$), which form the attachments of a straddling bridge.
• Figure 5: The bridge $Q$ has attachment set $A = S = \{s_1, s_2, s_3\}$. $S$ is straddled by $\Gamma$ with attachments $\{u, v\}$. If we explore all bridges of $\Pi$ attached to $u$, then $\Gamma$ will be naturally detected (top). Otherwise, we must have terminated early at some vertex $u_1$ before reaching $u$. \ref{['fact:local:candidate-precedes-U']} implies all bridges of $\Pi$ attached to some vertex $s_2$ were explored without terminating early. So $s_2$ must precede some unverified vertex $u_2$. This implies that $S$ is straddled by the unverified vertices (bottom).

Theorems & Definitions (105)

• Theorem 1.0
• Theorem 1.0
• definition 2.1: Volume
• definition 2.1: Balanced/Unbalanced $k$-Shredders
• Lemma 2.1
• definition 2.1: Capture
• Lemma 2.1
• Theorem 4.0: CT99
• definition 4.1: Bridge
• definition 4.2: Attachments