Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2210.16534

Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity

Jingyang Zhao·Mingyu Xiao
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Based on customer demand, we distinguish three variants of CVRP: unit-demand, splittable, and unsplittable. In this paper, we consider $k$-CVRP in general metrics and on general graphs, where $k$ is the vehicle capacity. All three versions are APX-hard for any fixed $k\geq3$. Assume that the approximation ratio of metric TSP is $\frac{3}{2}$. We present a $(\frac{5}{2}-\Theta(\frac{1}{\sqrt{k}}))$-approximation algorithm for the splittable and unit-demand cases, and a $(\frac{5}{2}+\ln2-\Theta(\frac{1}{\sqrt{k}}))$-approximation algorithm for the unsplittable case. Our approximation ratio is better than the previous results when $k$ is less than a sufficiently large value, approximately $1.7\times10^7$. For small values of $k$, we design independent and elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the approximation ratio from $1.792$ to $1.500$ for $k=3$, and from $1.750$ to $1.500$ for $k=4$. For the unsplittable case, we improve the approximation ratio from $1.792$ to $1.500$ for $k=3$, from $2.051$ to $1.750$ for $k=4$, and from $2.249$ to $2.157$ for $k=5$. The approximation ratio for $k=3$ surprisingly achieves the same value as in the splittable case. Our techniques, such as EX-ITP -- an extension of the classic ITP method, have the potential to improve algorithms for other routing problems as well.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jingyang Zhao, Mingyu Xiao. 2022-10-29. Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity. https://arxiv.org/abs/2210.16534

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

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗