Towards an efficient approximability for the Euclidean capacitated vehicle routing problem with time windows and multiple depots

We consider the Euclidean Capacitated Vehicle Routing Problem with Time Windows (CVRPTW). For the long time, approximability of this well-known problem in the class of algorithms with theoretical guarantees was poorly studied. This year, for the case of a single depot, we proposed two approximation algorithms, which are the Efficient Polynomial Time Approximation Schemes (EPTAS) for any fixed given capacity q and the number p of mutually disjunctive time windows. The former scheme extends the celebrated approach proposed by M. Haimovich and A. Rinnooy Kan and allows the evident parallelization, while the latter one has an improved time complexity bound and the enlarged domain in terms q = q(n) and p = p(n), where it retains polynomial time complexity. In this paper, we announce an extension of these results to the case of multiple depots. So, the first scheme is also EPTAS for any fixed parameters q, p, and m, where m is the number of depots, and remains PTAS for q = o(log log n) and mp = o(log log n). In other turn, the second one is a PTAS for p3q4 = O(log n) and (pq)2 log m = O(log n). © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Russian Foundation for Basic Research, RFBR: 17-08-01385, 19-07-01243Michaeffi Khachay was supported by the Russian Foundation for Basic Research, grants no. 17-08-01385 and 19-07-01243

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles with possibly different speeds, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot

The parsimonious property of cut covering problems and its applications

Includes bibliographical references (p. 26-27).Supported by a Presidential Young Investigator Award. DDM-91568118 Supported by Draper Laboratory.Dimitris Bertsimas and Chungpiaw Teo

Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems

We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general

Dial a Ride from k-forest

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying at most $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We prove that an $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & Raghavachari; our results give a different proof of a similar approximation guarantee--in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200

Choose Outsiders First: a mean 2-approximation random algorithm for covering problems

A high number of discrete optimization problems, including Vertex Cover, Set Cover or Feedback Vertex Set, can be unified into the class of covering problems. Several of them were shown to be inapproximable by deterministic algorithms. This article proposes a new random approach, called Choose Outsiders First, which consists in selecting randomly ele- ments which are excluded from the cover. We show that this approach leads to random outputs which mean size is at most twice the optimal solution.Comment: 8 pages The paper has been withdrawn due to an error in the proo

09261 Abstracts Collection -- Models and Algorithms for Optimization in Logistics

From June 21 to June 26, 2009 the Dagstuhl Seminar Perspectives Workshop 09261 ``Models and Algorithms for Optimization in Logistics \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available