Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2411.18312

Undirected 3-Fault Replacement Path in Nearly Cubic Time

Shucheng Chi·Ran Duan·Benyu Wang·Tianle Xie
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shucheng Chi, Ran Duan, Benyu Wang, Tianle Xie. 2024-11-27. Undirected 3-Fault Replacement Path in Nearly Cubic Time. https://arxiv.org/abs/2411.18312

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗