Low-Cost Arborescence Under Edge Faults

2026-05-13

AI Summary

This work addresses the problem of maintaining a minimum-cost arborescence in edge-weighted directed graphs under single-edge failures. It presents the first 2-approximate sparse fault-tolerant subgraph construction, combining arborescence algorithms, fault-tolerant subgraph techniques, and matroid theory to design a polynomial-time algorithm. The resulting subgraph has size $O(n^{3/2})$ and guarantees that, after any single-edge failure, a 2-approximation to the original minimum-cost arborescence can be recovered in $O(n^{3/2})$ time. Furthermore, within the matroid framework, the paper establishes a tight lower bound of $k \cdot \mathrm{rank}(E)$ on the size of any $k$-fault-tolerant preserver, thereby proving the theoretical optimality of the proposed construction for the single-failure case.

Abstract

Our input is a directed graph $G = (V,E)$ on $n$ vertices and $m$ edges with a designated root vertex $r$ and a function $cost: E \rightarrow \mathbb{R}_{\geq 0}$. The problem is to maintain a min-cost arborescence in $G$ in the presence of edge faults (a single fault at a time). Edge faults are transient and once the faulty edge is repaired, the original min-cost arborescence $\mathcal{T}$ is restored. Whenever an edge fault happens, we need to update $\mathcal{T}$ to a min-cost arborescence in $G-f$, where $f$ is the faulty edge. Since computing a min-cost arborescence in $G - f$ takes $O(m + n\log n)$ time, we seek to construct a sparse subgraph $H$ in a preprocessing step such that in the event of any edge $f$ failing, it suffices to compute a min-cost arborescence in $H - f$ in order to find a low-cost arborescence in $G - f$. In the unweighted setting, this is the fault-tolerant subgraph problem for single-source {\em reachability}. Baswana, Choudhary, and Roditty (SICOMP, 2018) showed a $k$-fault tolerant reachability subgraph of size $O(2^kn)$, where $k$ is the number of edge faults. We show a simple polynomial-time algorithm to construct a subgraph $H$ of size $O(n^{3/2})$ such that, for any $f \in E$, a min-cost arborescence in $H-f$ is a 2-approximation of a min-cost arborescence in $G-f$. Thus whenever an edge fault happens, we can find a 2-approximate min-cost arborescence in $G-f$ in $O(n^{3/2})$ time. Our second problem is in the matroid setting. The input is a matroid $M = (E, {\cal I})$ with a function $cost: E \rightarrow \mathbb{R}$. The problem is to compute a sparse $S \subseteq E$ (called a $k$-fault tolerant preserver) such that for any $F \subseteq E$ with $|F| \le k$, the matroid $M|(S\setminus F)$ contains a min-cost basis of $M|(E\setminus F)$. We show a tight bound of $k.rank(E)$ on the size of a $k$-fault tolerant preserver.

Problem

• arborescence
• edge faults
• fault-tolerant subgraph
• matroid
• min-cost basis

Innovation

• fault-tolerant arborescence
• sparse subgraph
• approximation algorithm
• matroid preserver
• edge failure