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

Jayaprakash, Aditya; Salavatipour, Mohammad R.

doi:10.1137/1.9781611977073.37

Cited by 7 publications

(7 citation statements)

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“…Becker and Paul [BP19] gave a bicriteria polynomial time approximation scheme, allowing the tour capacity to be violated by an fraction. Jayaprakash and Salavatipour [JS22] gave a quasi-polynomial time approximation scheme (QPTAS). A polynomial time approximation scheme (PTAS) was given by Mathieu and Zhou [MZ21].…”

Section: Related Workmentioning

confidence: 99%

“…There have been polynomial time constant-factor approximation algorithms for general metrics [HR85, AG90, BDO06, BTV21]. QPTAS algorithms have been designed for the unit demand CVRP in several metrics: Euclidean [DM15], trees or bounded treewidth and beyond [JS22], planar and bounded-genus graphs with fixed tour capacity [BKS17], and minor-free graphs [CFKL20]. When the tour capacity k is small, the unit demand CVRP admits PTAS algorithms on several metrics: Euclidean [HR85, AKTT97, ACL10], planar graphs [BKS19], graphs of bounded highway dimension [BKS18], bounded genus graphs and bounded treewidth graphs [CFKL20].…”

Section: Related Workmentioning

confidence: 99%

“…In order to achieve the third property of the claim, we construct S * by modifying Ŝ * . To that end, we use the technique of the adaptive rounding, which has previously been used in the unitdemand setting to design a QPTAS [JS22] and a PTAS [MZ21] for the tree CVRP. The construction of S * is identical to the construction of OPT Combining, we have cost( Ŝ * ) < (3/2 + 7 ) • opt.…”

Section: Analysis On the Cost Ofmentioning

confidence: 99%

“…. , en, such that the weight of e1 equals 1 and the weight of ei equals 0 for each i ∈ [2, n].2 The unsplittable path CVRP was called the train delivery problem in [DMM10, CGHI13, Rot12].3 A technical detail is that both[JS22] and[MZ21] have assumptions on the tour capacity k barring it from being exponential in n.…”

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confidence: 99%

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A Tight $(\frac{3}{2}+ε)$-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

Mathieu¹,

Zhou²

2022

Preprint

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Analysis On the Cost Ofmentioning

confidence: 99%

mentioning

confidence: 99%

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A Tight $(\frac{3}{2}+ε)$-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

Mathieu¹,

Zhou²

2022

Preprint

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“…When the capacity Q is a part of the input, a QPTAS for VRP in Euclidean space of constant dimension is known [DM15,ACL09]; it remains a major open problem to design a PTAS for VRP even for the Euclidean plane. For trees, a PTAS was only obtained by a recent work of Mathieu and Zhou [MZ21], improving upon the QPTAS of Jayaprakash and Salavatipour [JS22]. No PTAS is known beyond trees, such as planar graphs or bounded treewidth graphs.…”

Section: Introductionmentioning

confidence: 99%

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Filtser¹,

Lê²

2022

Preprint

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Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minorfree graphs of diameter D into graphs of treewidth O (log n) with expected additive distortion + D. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to O (log log n) 2 . As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from n O (log(n)) to n O (log log(n)) 3 . Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter D into graphs of treewidth O((log log n) 2 ) ) and additive distortion + D that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time.A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth O(log log n) and hop-diameter O(log log n). This result may be of independent interest.

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Improved Approximations for Capacitated Vehicle Routing with Unsplittable Client Demands

Friggstad

Mousavi

Rahgoshay

et al. 2022

Integer Programming and Combinatorial Optimization

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

Cited by 7 publications

References 0 publications

A Tight $(\frac{3}{2}+ε)$-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

A Tight $(\frac{3}{2}+ε)$-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Improved Approximations for Capacitated Vehicle Routing with Unsplittable Client Demands

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