The Steiner Ratio Gilbert–Pollak Conjecture Is Still Open

Ivanov, A O; Tuzhilin, A. A.

doi:10.1007/s00453-011-9508-3

Cited by 27 publications

(14 citation statements)

References 7 publications

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“…If the distance function is Euclidean, κ is conjectured to be 2 √ 3 ≈ 1.1547. This conjecture is believed to be true, but is still open and unproven [12]. If it holds, then for the Euclidean case CDP-T is a 2 √ 3 -approximation to CDP-TA.…”

Section: Transfers At Any Locationmentioning

confidence: 97%

Optimizing for Transfers in a Multi-vehicle Collection and Delivery Problem

Coltin

Veloso

2014

Springer Tracts in Advanced Robotics

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We address the Collection and Delivery Problem (CDP) with multiple vehicles, such that each collects a set of items at different locations and delivers them to a dropoff point. The goal is to minimize either delivery time or the total distance traveled. We introduce an extension to the CDP: what if a vehicle can transfer items to another vehicle before making the final delivery? By dividing the labor among multiple vehicles, the delivery time and cost may be reduced. However, introducing transfers increases the number of feasible schedules exponentially. In this paper, we investigate this Collection and Delivery Problem with Transfers (CDP-T), discuss its theoretical underpinnings, and introduce a two-approximate polynomial time algorithm to minimize total distance travelled. Furthermore, we show that allowing transfers to take place at any location for the CDP-T results in at most a factor of two improvement. We demonstrate our approximation algorithms on large simulated problem instances. Finally, we deploy our algorithms on robots that transfer and deliver items autonomously in an office building.

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Section: Transfers At Any Locationmentioning

confidence: 97%

Optimizing for Transfers in a Multi-vehicle Collection and Delivery Problem

Coltin

Veloso

2014

Springer Tracts in Advanced Robotics

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“…M ST (si) is the minimum spanning tree connecting si and all its destinations. According to a series of theoretical studies [20], it is often believed that ||Ti|| ≥ β0 · ||M ST (si)||, where β0 is constant. The following lemma gives a lower bound of ||M ST (si)||.…”

Section: Competitive Intensitymentioning

confidence: 99%

Capacity of wireless networks with multiple types of multicast sessions

Zheng

Gao

et al. 2014

Proceedings of the 15th ACM International Symposium on Mobile Ad Hoc Networking and Computing

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A thorough understanding of capacity of wireless networks can help with effective design and efficient employment of wireless networks. Much effort has been spent on investigating capacity of multicast which is a popular communication model and generalization of unicast and broadcast. However, most previous works assume homogeneous traffic patterns, which is not meaningful for practical applications. This paper analyzes the capacity of wireless networks with multiple types of multicast sessions without the assumption of homogeneous traffic patterns. A new network model is proposed accommodating practical traffic patterns and the capacity is analyzed accordingly. A theoretical upper bound is derived, and a feasible transmission scheme with capacity lower bound is presented. Two bounds are asymptotically tight, that is, in the order of Θ(where a is the side length of the deployed region, r is the transmission range, ns is the number of multicast sessions, and ki and Ri are parameters of multicast session i. Furthermore, the variation of capacity towards different numbers of and distributions of destinations is illustrated.

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“…Remove edge e from T and consider the two connected components (subtrees) T 1 and T 2 . Let t i and t j be the terminal endpoints of edge e. In N T there is a unique 11 Here we give a brief history of the Steiner ratio conjecture, based largely on the 2012 account by Ivanov and Tuzhilin [220], and their update in 2014 [221]. In 1992 a paper by Du and Hwang [135] presented a proof of the Steiner ratio conjecture.…”

Section: Lemma 17 ([106] Lower Bound For Steiner Ratio)mentioning

confidence: 99%