An Extension of the Das and Mathieu QPTAS to the Case of Polylog Capacity Constrained CVRP in Metric Spaces of a Fixed Doubling Dimension

Khachay, Michael; Ogorodnikov, Yuri; Khachay, Daniel

doi:10.1007/978-3-030-49988-4_4

Cited by 6 publications

(2 citation statements)

References 30 publications

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“…For high dimensional Euclidean spaces R d , Khachay et al [KD16] showed a PTAS when Q is O(log 1/d n). For graphs of bounded doubling dimension, Khachay et al [KO20] gave a QPTAS when the optimal number of tours is polylog(n) and Khachay et al [KOK20] gave a QPTAS when Q is polylog(n).…”

Section: Related Workmentioning

confidence: 99%

Improved Approximations for CVRP with Unsplittable Demands

Friggstad¹,

Mousavi²,

Rahgoshay³

et al. 2021

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In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics. In CVRP, introduced by Dantzig and Ramser (1959), we are given a set of points (clients) V together with a depot r in a metric space, with each v ∈ V having a demand d v > 0, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle, each starting and ending at the depot, such that each client is visited at least once and the total demands of the clients in each tour is at most Q. In the unsplittable variant we study, the demand of a node must be served entirely by one tour. We present two approximation algorithms for unsplittable CVRP: a combinatorial (α + 1.75)-approximation, where α is the approximation factor for the Traveling Salesman Problem, and an approximation algorithm based on LP rounding with approximation guarantee α + ln(2) + δ ≈ 3.194 + δ in n O(1/δ) time. Both approximations can further be improved by a small amount when combined with recent work by Blauth, Traub, and Vygen (2021), who obtained an (α + 2 • (1 − ε))-approximation for unsplittable CVRP for some constant ε depending on α (ε > 1/3000 for α = 1.5).

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Section: Related Workmentioning

confidence: 99%

Improved Approximations for CVRP with Unsplittable Demands

Friggstad¹,

Mousavi²,

Rahgoshay³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For high dimensional Euclidean spaces R d , Khachay et al [20] showed a PTAS when Q is O(log 1/d n). For graphs of bounded doubling dimension, Khachay et al [21] gave a QPTAS when the number of tours is polylog(n) and Khachay et al [22] gave a QPTAS when Q is polylog(n).…”

Section: Related Workmentioning

confidence: 99%

Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension

Jayaprakash¹,

Salavatipour²

2021

Preprint

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In this paper we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [14], we are given a graph G = (V, E) with metric edges costs, a depot r ∈ V , and a vehicle of bounded capacity Q. The goal is to find minimum cost collection of tours for the vehicle that return to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting each node v has a demand dv and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tour (splittable). The best known approximation algorithm for general graphs has ratio α + 2(1 − ) (for the unsplittable) and α + 1 − (for the splittable) for some fixed > 1 3000 , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [15] presented an approximation scheme with time n log O(1/ ) n for Euclidean plane R 2 . No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this paper we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for Euclidean plane with run time n O(log 10 n/ 9 ) .

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