Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.19537

Approximation Algorithms for Emergency Vertex Cover with Few Stations and Fault-Tolerant Coverage

Quanlong Niu·Riheng Jia
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Emergency Vertex Cover problem (Em-VC) assigns budgets to stations at graph vertices to cover every edge while minimizing the total budget. The cost of covering an edge from a station is the shortest-path distance from the station to the nearer endpoint plus the edge length. We study how restricting the number of stations affects the cost of ordinary and fault-tolerant coverage. In ordinary Em-VC, we prove that the optimal solution using at most two stations costs at most $5/4$ times the unrestricted optimum, improving the known $4/3$ guarantee for a single station. A recurrence extends this bound to larger station limits, giving approximation factors that converge to $\sqrt{5}-1$. Budget-threshold enumeration computes the restricted optimum in polynomial time for every fixed station limit. Under $k$-fold coverage, every edge must be covered by at least $k$ distinct stations, ensuring coverage after any $k-1$ station failures. We prove that selecting the $k$ vertices with the smallest full-coverage costs yields a $2$-approximation relative to the unrestricted optimum for $k$-fold coverage. When $k=2$, we combine upper bounds derived from the structure of the covering balls to improve the approximation guarantee to $5/3$.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Quanlong Niu, Riheng Jia. 2026-10-14. Approximation Algorithms for Emergency Vertex Cover with Few Stations and Fault-Tolerant Coverage. https://arxiv.org/abs/2610.19537

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

KEEP EXPLORING

Related papers

arXiv2026

O(1) Expected Insertion for Random Walk d-ary Cuckoo Hashing up to the Load Threshold

The random walk $d$-ary cuckoo hashing algorithm was defined by Fotakis, Pagh, Sanders, and Spirakis to generalize and improve upon the standard cuckoo hashing algorithm of Pagh and Rodler. Random walk $d$-ary cuckoo hashing has low space overhead, guaranteed fast access, and fast in practice insertion time. In this paper, we give a theoretical insertion time bound for this algorithm. More precisely, for every $d\ge 3$ random hashes, let $c_d^*$ be the sharp threshold for the load factor at which a valid assignment of $cm$ objects to a hash table of size $m$ exists with high probability. We show that for any $d\ge 3$ hashes and load factor $c<c_d^*$, the expectation of the random walk insertion time is $O(1)$, that is, a constant depending only on $d$ and $c$ but not $m$.

cs.DS↗
arXiv2026

Approximating Unsplittable Capacitated Vehicle Routing: Analysis via Factor-Revealing LPs

The capacitated vehicle routing problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Given a depot and a set of customers with demands in a metric space, the objective is to find a set of tours, each starting and ending at the depot, operated by a capacitated vehicle located at the depot, such that all customers are served and the total travel cost is minimized. We consider the unsplittable variant, where the demand of each customer must be served entirely by a single tour. The previous best approximation ratio was about $α+1.694$, where $α$ is the approximation ratio for the traveling salesman problem. We propose improved approximation algorithms. First, we introduce a unified algorithmic framework that generalizes and integrates known linear programming (LP) rounding and matching-based approaches. We show that these techniques are complementary and can be combined to yield improved approximation ratios. Second, we develop a novel analytical framework based on customer demand discretization, which yields new structural bounds and, through factor-revealing LPs, achieves expected approximation ratios of about $α+1.647$ for the general-capacity case and $α+1.548$ for the fixed-capacity case. Finally, we derandomize our algorithms using a priori 2-tour sets and the method of conditional expectations while maintaining nearly the same approximation ratios.

cs.DS↗
arXiv2026

Entrywise Low-Rank Approximation and Matrix $p \rightarrow q$ Norms via Global Correlation Rounding

Given a matrix $A$, the goal of the entrywise low-rank approximation problem is to find $\operatorname{argmin} \|A-B\|_p$ over all rank-$k$ matrices $B$, where $\| \cdot \|_p$ is the entrywise $\ell_p$ norm. When $p = 2$ this well-studied problem is solved by the singular value decomposition, but for $p \neq 2$ the problem becomes computationally challenging. For every even $p > 2$ and every fixed $k$, we give the first polynomial-time approximation scheme for this problem, improving on the $(3 + \varepsilon)$ approximation of Ban, Bhattiprolu, Bringmann, Kolev, Lee, and Woodruff, the bi-criteria approximation of Woodruff and Yasuda, and the additive approximation scheme of Anderson, Bakshi, and Hopkins. Prior algorithmic approaches based on sketching and column selection, which yielded a polynomial-time approximation scheme in the $p < 2$ setting, face concrete barriers when $p > 2$. Instead, we use the Sherali-Adams hierarchy of convex programs, and in so doing suggest a methodology for how to use convex hierarchies to design polynomial-time approximation schemes for continuous optimization problems. We use the same algorithmic strategy to give a new family of additive approximation algorithms for matrix $p \rightarrow q$ norms, which are intimately related to small-set expansion and quantum information. In particular, we give the first nontrivial additive approximation algorithms in the regime $p < 2 < q$.

cs.DS↗