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A PTAS for Capacitated Vehicle Routing on Trees [article]

Claire Mathieu, Hang Zhou
2022 arXiv   pre-print
We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity.  ...  There is an approximation scheme for the unit demand capacitated vehicle routing problem (CVRP) on trees with polynomial running time. Corollary 2.  ...  There is an approximation scheme for the splittable capacitated vehicle routing problem (CVRP) on trees with running time polynomial in the number of vertices n and the tour capacity k.  ... 
arXiv:2111.03735v3 fatcat:wwu5hh7eavewlcvr6kg7yp5tru

A PTAS for Capacitated Vehicle Routing on Trees

Claire Mathieu, Hang Zhou, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruff
2022
We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity.  ...  If b i is a I C A L P 2 0 2 PTAS for Capacitated Vehicle Routing on Trees subtour at r c for some component c, then r c must belong to the tour a; and if b i is a group of pieces in some component c, then  ...  The proof of Theorem 13 contains several novelties in our I C A L P 2 0 2 2 95:6 A PTAS for Capacitated Vehicle Routing on Trees v 1 v 2 v 3 v 4 v 5 Figure 3 At each vertex of the tree, the dynamic program  ... 
doi:10.4230/lipics.icalp.2022.95 fatcat:ve6z3ijihfhy5ppgfxvcuuj2by

A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs [article]

Amariah Becker and Philip N. Klein and Aaron Schild
2019 arXiv   pre-print
The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity  ...  In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded.  ...  Related Work Capacitated Vehicle Routing There is a substantial body of work on approximation algorithms for Capacitated Vehicle Routing.  ... 
arXiv:1901.07032v1 fatcat:qbdtm7gbangy5e7fsnq3grkkem

Polynomial-Time Approximation Schemes for k-center, k-median, and Capacitated Vehicle Routing in Bounded Highway Dimension

Amariah Becker, Philip N. Klein, David Saulpic, Michael Wagner
2018 European Symposium on Algorithms  
This result implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing: the tours can go from one depot to another.  ...  We show that this embedding yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension.  ...  Acknowledgements Thanks to Andreas Feldmann and Vincent Cohen-Addad for helpful discussions and comments.  ... 
doi:10.4230/lipics.esa.2018.8 dblp:conf/esa/BeckerKS18 fatcat:tbquhn36jzhajdt7u36ti3mo5m

A Framework for Vehicle Routing Approximation Schemes in Trees [article]

Amariah Becker, Alice Paul
2019 arXiv   pre-print
We develop a general framework for designing polynomial-time approximation schemes (PTASs) for various vehicle routing problems in trees.  ...  This simplification relies on partitioning the tree into clusters such that there exists a near-optimal solution in which every vehicle that visits a given cluster takes on one of a few forms.  ...  Given an instance of Capacitated Vehicle Routing on a tree, if there exists a solution of total length k and capacity Q, then for any > 0, there is a polynomial-time algorithm that finds a solution of  ... 
arXiv:1807.04308v2 fatcat:d73dcoho3zffnhu3gvf4i66z34

Polynomial-Time Approximation Schemes for k-Center and Bounded-Capacity Vehicle Routing in Graphs with Bounded Highway Dimension [article]

Amariah Becker, Philip N. Klein, David Saulpic
2017 arXiv   pre-print
This embedding result implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing: the tours can go from one depot to another.  ...  We show that this theorem yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension.  ...  Acknowledgements Thanks to Andreas Feldmann and Vincent Cohen-Addad for helpful discussions and comments.  ... 
arXiv:1707.08270v5 fatcat:emz5iemeo5dcpb3ggwfei3vn5e

A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees [article]

Amariah Becker
2018 arXiv   pre-print
In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of (√(41)-1)/4 by Asano et al., while using  ...  Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded  ...  A natural question is whether better performance can be achieved for restricted graph classes. One line of research has focused on approximating Capacitated Vehicle Routing in trees.  ... 
arXiv:1804.08791v1 fatcat:v5ns5vjeuvejbcz3f2dzy5xgnq

Approximation Algorithms for Capacitated Location Routing

Tobias Harks, Felix G König, Jannik Matuschke
2013 Transportation Science  
We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances.  ...  vehicle routing by this framework.  ...  Acknowledgments The authors would like to thank two anonymous referees, the guest editor, and the editor in chief for motivating us to conduct the computational study in §6.  ... 
doi:10.1287/trsc.1120.0423 fatcat:i5nine375bf45lywskbd2y346q

Improved Approximations for CVRP with Unsplittable Demands [article]

Zachary Friggstad, Ramin Mousavi, Mirmahdi Rahgoshay, Mohammad R. Salavatipour
2021 arXiv   pre-print
In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics.  ...  some constant ϵ depending on α (ϵ > 1/3000 for α = 1.5).  ...  Also, we treat r as a separate node from the rest of the nodes. Formally: Definition 3 (CAPACITATED VEHICLE ROUTING).  ... 
arXiv:2111.08138v1 fatcat:2yxr6x62i5cl3cd3smj3tvz7iy

PTAS for k-tour cover problem on the plane for moderately large values of k [article]

Anna Adamaszek, Artur Czumaj, Andrzej Lingas
2009 arXiv   pre-print
In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends  ...  We significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k <= 2^log^δn, where δ = δ(ϵ).  ...  The k-TC problem (capacitated vehicle routing problem) is one of the central special cases of a more general vehicle routing problem, introduced by Dantzig and Ramser [6] fifty years ago, and studied  ... 
arXiv:0904.2576v1 fatcat:mch4lnorqvaenbx65sr2q7qu3i

A Polynomial-Time Exact Algorithm for k -Depot Capacitated Vehicle Routing Problem on a Tree

Liang Xu, Tong Zhang, Rantao Hu, Jialing Guo, Lu Zhen
2021 Mathematical Problems in Engineering  
This paper proposes a polynomial-time exact algorithm for the k -depot capacitated vehicle routing problem on a tree for fixed k ( k -depot CVRPT for short), which involves dispatching a fixed number of  ...  k capacitated vehicles in depots on a tree-shaped graph to serve customers with the objective of minimizing total distance traveled.  ...  Introduction is article investigates routing k capacitated vehicles in fixed depots to serve customers.  ... 
doi:10.1155/2021/6648044 fatcat:q2apjlkl5jeyjb7o3e5sskv3r4

A Constant Factor Approximation for Capacitated Min-Max Tree Cover [article]

Syamantak Das, Lavina Jain, Nikhil Kumar
2019 arXiv   pre-print
We give the first constant factor approximation for the hard uniform capacitated version of this problem, where, an input parameter λ upper bounds the number of vertices that can be covered by any of the  ...  Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al., OR Letters 2004]  ...  As mentioned in Section 1, there is a PTAS known for the uncapacitated rooted MMTC problem on a tree metric.  ... 
arXiv:1907.08304v2 fatcat:5rqcnn62nzbmjhwhq7zoteizdm

Improving the Approximation Ratio for Capacitated Vehicle Routing [article]

Jannis Blauth, Vera Traub, Jens Vygen
2020 arXiv   pre-print
We devise a new approximation algorithm for capacitated vehicle routing.  ...  Our algorithm yields a better approximation ratio for general capacitated vehicle routing as well as for the unit-demand case and the splittable variant. Our results hold in arbitrary metric spaces.  ...  x(δ(A)) is bounded by a constant (these can be enumerated in polynomial time by [23] ).  ... 
arXiv:2011.05235v1 fatcat:7rqx2zn3gjhirfz2nuydr4gabi

Approximating Airports and Railways

Anna Adamaszek, Antonios Antoniadis, Amit Kumar, Tobias Mömke, Marc Herbstritt
2018 Symposium on Theoretical Aspects of Computer Science  
We obtain the first bicriteria approximation algorithm for AR for the general metric case, which yields a 4-approximate solution with a resource augmentation of the airport capacity k by a factor of 2.  ...  An instance of the airport and railway problem consists of a set of points in the corresponding metric, together with a non-negative weight for each point, and a parameter k.  ...  This problem is of particular interest because it models the Capacitated Vehicle Routing Problem (CVRP).  ... 
doi:10.4230/lipics.stacs.2018.5 dblp:conf/stacs/AdamaszekA0M18 fatcat:4dzilucnhfbwxkdlzie75ndqwu

A Constant Factor Approximation for Capacitated Min-Max Tree Cover

Syamantak Das, Lavina Jain, Nikhil Kumar, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
In the capacitated version, there is a hard uniform capacity (λ) on the number of vertices a tree can cover.  ...  tree is a subgraph of G.  ...  As mentioned in Section 1, there is a PTAS known for the uncapacitated rooted MMTC problem on a tree metric.  ... 
doi:10.4230/lipics.approx/random.2020.55 dblp:conf/approx/DasJK20 fatcat:tox7l32lmfeknb3d4b76dq73ui
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