arXiv subjects

Benyu Wang

Publications and source records attributed to Benyu Wang.

5 recordsLinked to original sources

arXiv2026

Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition

We give an improved connectivity oracle under vertex failures. After a set of $k$ vertices fails, our oracle performs an $O(k^{6})$-time update independent of the graph size $n$, and then answers pairwise connectivity queries in optimal $O(k)$ time. For constant $k$, it uses near-linear space and can be built in near-linear preprocessing time. In contrast, all prior oracles with $n$-independent update time[PSS+22, vdBS19] either require $\Omega(n^{2})$ space or incur $2^{2^{O(k)}}$ update and query time. Moreover, their preprocessing time is polynomially large in $n$, far from near-linear. Our oracle builds on the unbreakable decomposition framework of[PSS+22], but introduces three new ingredients: (i) shortcutting over the tree decomposition to reduce space from quadratic to near-linear, (ii) bootstrapping that leverages $n$-dependent oracles internally to obtain near-linear preprocessing, and (iii) a new patch set mechanism that yields conditionally optimal $O(k)$ query time.

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arXiv2025

Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies

We give almost-linear-time algorithms for approximating rooted minimum cut and maximum arborescence packing in directed graphs, two problems that are dual to each other [Edm73]. More specifically, for an $n$-vertex, $m$-edge directed graph $G$ whose $s$-rooted minimum cut value is $k$, our first algorithm computes an $s$-rooted cut of size at most $O(k\log^{5} n)$ in $m^{1+o(1)}$ time, and our second algorithm packs $k$ $s$-rooted arborescences with $n^{o(1)}$ congestion in $m^{1+o(1)}$ time, certifying that the $s$-rooted minimum cut is at least $k / n^{o(1)}$. Our first algorithm also works for weighted graphs. Prior to our work, the fastest algorithms for computing the $s$-rooted minimum cut were exact but had super-linear running time: either $\tilde{O}(mk)$ [Gab91] or $\tilde{O}(m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\})$ [CLN+22]. The fastest known algorithms for packing $s$-rooted arborescences had no congestion, but required $\tilde{O}(m \cdot \mathrm{poly}(k))$ time [BHKP08].

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arXiv2025

Near-Optimal Fault-Tolerant Strong Connectivity Preservers

A $k$-fault-tolerant connectivity preserver of a directed $n$-vertex graph $G$ is a subgraph $H$ such that, for any edge set $F \subseteq E(G)$ of size $|F| \le k$, the strongly connected components of $G - F$ and $H - F$ are the same. While some graphs require a preserver with $\Omega(2^{k}n)$ edges [BCR18], the best-known upper bound is $\tilde{O}(k2^{k}n^{2-1/k})$ edges [CC20], leaving a significant gap of $\Omega(n^{1-1/k})$. In contrast, there is no gap in undirected graphs; the optimal bound of $\Theta(kn)$ has been well-established since the 90s [NI92]. We nearly close the gap for directed graphs; we prove that there exists a $k$-fault-tolerant connectivity preserver with $O(k4^{k}n\log n)$ edges, and we can construct one with $O(8^{k}n\log^{5/2}n)$ edges in $\text{poly}(2^{k}n)$ time. Our results also improve the state-of-the-art for a closely related object; a \textit{$k$-connectivity preserver} of $G$ is a subgraph $H$ where, for all $i \le k$, the strongly $i$-connected components of $G$ and $H$ agree. By a known reduction, we obtain a $k$-connectivity preserver with $O(k4^{k}n\log n)$ edges, improving the previous best bound of $\tilde{O}(k2^{k}n^{2-1/(k-1)})$ [CC20]. Therefore, for any constant $k$, our results are optimal to a $\log n$ factor for both problems. Lastly, we show that the exponential dependency on $k$ is not inherent for $k$-connectivity preservers by presenting another construction with $O(n \sqrt{kn})$ edges.

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arXiv2024

Undirected 3-Fault Replacement Path in Nearly Cubic Time

Given a graph $G=(V,E)$ and two vertices $s,t\in V$, the $f$-fault replacement path ($f$FRP) problem computes for every set of edges $F$ where $|F|\leq f$, the distance from $s$ to $t$ when edges in $F$ fail. A recent result shows that 2FRP in directed graphs can be solved in $\tilde{O}(n^3)$ time [arXiv:2209.07016]. In this paper, we show a 3FRP algorithm in deterministic $\tilde{O}(n^3)$ time for undirected weighted graphs, which almost matches the size of the output. This implies that $f$FRP in undirected graphs can be solved in almost optimal $\tilde{O}(n^f)$ time for all $f\geq 3$. To construct our 3FRP algorithm, we introduce an incremental distance sensitivity oracle (DSO) with $\tilde{O}(n^2)$ worst-case update time, while preprocessing time, space, and query time are still $\tilde{O}(n^3)$, $\tilde{O}(n^2)$ and $\tilde{O}(1)$, respectively, which match the static DSO [Bernstein and Karger 2009]. Here in a DSO, we can preprocess a graph so that the distance between any pair of vertices given any failed edge can be answered efficiently. From the recent result in [arXiv:2211.05178], we can obtain an offline dynamic DSO from the incremental worst-case DSO, which makes the construction of our 3FRP algorithm more convenient. By the offline dynamic DSO, we can also construct a 2-fault single-source replacement path (2-fault SSRP) algorithm in $\tilde{O}(n^3)$ time, that is, from a given vertex $s$, we want to find the distance to any vertex $t$ when any pair of edges fail. Thus the $\tilde{O}(n^3)$ time complexity for 2-fault SSRP is also almost optimal. Now we know that in undirected graphs 1FRP can be solved in $\tilde{O}(m)$ time [Nardelli, Proietti, Widmayer 2001], and 2FRP and 3FRP in undirected graphs can be solved in $\tilde{O}(n^3)$ time. In this paper, we also show that a truly subcubic algorithm for 2FRP in undirected graphs does not exist under APSP-hardness conjecture.

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arXiv2023

Tight Conditional Lower Bounds for Vertex Connectivity Problems

We study the fine-grained complexity of graph connectivity problems in unweighted undirected graphs. Recent development shows that all variants of edge connectivity problems, including single-source-single-sink, global, Steiner, single-source, and all-pairs connectivity, are solvable in $m^{1+o(1)}$ time, collapsing the complexity of these problems into the almost-linear-time regime. While, historically, vertex connectivity has been much harder, the recent results showed that both single-source-single-sink and global vertex connectivity can be solved in $m^{1+o(1)}$ time, raising the hope of putting all variants of vertex connectivity problems into the almost-linear-time regime too. We show that this hope is impossible, assuming conjectures on finding 4-cliques. Moreover, we essentially settle the complexity landscape by giving tight bounds for combinatorial algorithms in dense graphs. There are three separate regimes: (1) all-pairs and Steiner vertex connectivity have complexity $\hatΘ(n^{4})$, (2) single-source vertex connectivity has complexity $\hatΘ(n^{3})$, and (3) single-source-single-sink and global vertex connectivity have complexity $\hatΘ(n^{2})$. For graphs with general density, we obtain tight bounds of $\hatΘ(m^{2})$, $\hatΘ(m^{1.5})$, $\hatΘ(m)$, respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time.

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