Towards an efficient approximability for the Euclidean Capacitated Vehicle Routing Problem with Time Windows and multiple depots

Michael Khachay, Yuri Ogorodnikov
2019 IFAC-PapersOnLine
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
more » ... lebrated 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 p 3 q 4 = O(log n) and (pq) 2 log m = O(log n). In this paper, we consider the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) (Kumar and Panneerselvam, 2012; Toth and Vigo, 2014) , which is an extension of the CVRP, where each customer should be serviced at a specified time interval, called a time win-dow. There are two different types of time windows. For the former type, called hard, any feasible solution should visit each customer within its dedicated time window exactly, whilst, for the latter one, known as soft, this constraint can be violated barring some penalty cost. Along with traditional applications in postal or bank deliveries, industrial refuse collection, school bus scheduling, etc., CVRPTW with hard windows (or just CVRPTW) is a widely employed mathematical model in continent-scale distribution of building materials (Pace et al., 2015) , in the low-carbon economy (Shen et al., 2018) , in dial-ride company planning (Gschwind and Irnich, 2015) and other practical transportation problems (see, e.g. (Savelsbergh and van Woensel, 2016)). In this paper, our goal is to propose for CVRPTW novel efficient algorithms with theoretical performance guarantees. Therefore, despite the obvious progress in solving practical instances of this problem by local-search heuristics (Hashimoto and Yagiura, 2008), genetic (Vidal et al., 2013) , memetic (Blocho and Czech, 2013; Nalepa and Blocho, 2016) , and ant colony algorithms (Necula et al., 2017) , in the following short literature overview, we concentrate intensionally on results concerning the complexity of CVRPTW and its approximability with theoretically proven bounds. Abstract: 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 p 3 q 4 = O(log n) and (pq) 2 log m = O(log n). Abstract: 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 p 3 q 4 = O(log n) and (pq) 2 log m = O(log n). PTAS for q = o(log log n) and mp = o(log log n). In other turn, the second one is a PTAS for p 3 q 4 = O(log n) and (pq) 2 log m = O(log n). Abstract: 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 p 3 q 4 = O(log n) and (pq) 2 log m = O(log n).