The Multi-source Beachcombers’ Problem

Czyzowicz, Jurek; Gąsieniec, Leszek; Georgiou, Konstantinos; Kranakis, Evangelos; MacQuarrie, Fraser

doi:10.1007/978-3-662-46018-4_1

Cited by 5 publications

(26 citation statements)

References 16 publications

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“…Our results for the cycle topology provide a partial answer to an open question posed in [11] and [12], concerning the study of the problem in different domain topologies.…”

Section: Introductionmentioning

confidence: 66%

“…We show that the structure of the optimal solutions to the single-source Beachcombers' Problem on a cycle is identical to the structure of the optimal solutions to the two-source Beachcombers' Problem on a finite segment, as defined in [12]. This implies that the results from [12] for the case of two distinct sources are carried over to the case of the cycle, yielding an NP-hardness result as well as the existence of efficient approximation algorithms for the problem. In particular, the NP-hardness of the single-source Beachcombers' Problem on the cycle seems at first somewhat surprising, in view of the existence of an efficient algorithm generating optimal schedules for the single-source problem on a finite segment.…”

Section: Introductionmentioning

confidence: 93%

“…The analogy to beachcombers has been introduced in [11] to bring out that, e.g., a beachcomber looking for things of value performs a meticulous search of the beach, which takes significantly more time than simply walking from one point of the beach to another. Further motivation for the two-speed model can be found in [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, a 2-competitive (online) algorithm was presented for the case where all robots are initally collocated on the origin of a semi-infinite line. In [12], the Beachcombers' Problem was studied for the case of more than one starting positions on a finite segment of known length. For a fixed number t ≥ 2 of starting positions, the t-source Beachcombers' Problem on a finite segment asks to find t starting points on the segment, an assignment of the robots to the starting points, and a search schedule which concludes the search of the finite segment as quickly as possible.…”

Section: Introductionmentioning

confidence: 99%

“…For a fixed number t ≥ 2 of starting positions, the t-source Beachcombers' Problem on a finite segment asks to find t starting points on the segment, an assignment of the robots to the starting points, and a search schedule which concludes the search of the finite segment as quickly as possible. It was shown in [12] that this problem is NP-hard for t = 2, even when all robots have the same walking speed, that the optimal solution can be computed efficiently when all robots have the same searching speed, and that there exist a deterministic approximation algorithm for t = 2 and a randomized approximation algorithm for general t.…”

Section: Introductionmentioning

confidence: 99%

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Abstract. A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We first consider the online Beachcombers' Problem, where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as, where tA(ℓ) denotes the time when the search of the segment [0, ℓ] is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over ℓ ∈ N * or ℓ ≥ 1. We prove that the LeapFrog algorithm, which was proposed in [Czyzowicz et al., SIROCCO 2014, LNCS 8576, pp. 23-36 (2014], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting. For the offline version of the Beachcombers' Problem, we consider the single-source Beachcombers' Problem on the cycle, as well as the multisource Beachcombers' Problem on the cycle and on the finite segment. For the single-source Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the two-source Beachcombers' Problem on a finite segment. In consequence, by using results from [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3-21 (2014], we prove that the single-source Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the multi-source variant of the Beachcombers' Problem on the cycle and on the finite segment, we obtain efficient approximation algorithms.

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“…Our results for the cycle topology provide a partial answer to an open question posed in [11] and [12], concerning the study of the problem in different domain topologies.…”

Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 93%