Improved integrality gap upper bounds for traveling salesperson problems with distances one and two

Mnich, Matthias; Mömke, Tobias

doi:10.1016/j.ejor.2017.09.036

Cited by 4 publications

(2 citation statements)

References 22 publications

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“…As such, much of the work in the literature has been on more specific distance functions. Some notable examples include graphic TSP [13,20,21,25,8] where the distances are the shortest paths over an arbitrary unweighted undirected graph, (1, 2)-TSP [1,8,5,16,19] where the distances are 1 or 2, and more generally metric TSP [15,12,10,26] where the distances satsify triangle inequality.…”

Section: Introductionmentioning

confidence: 99%

Sublinear Algorithms for TSP via Path Covers

Behnezhad¹,

Roghani²,

Rubinstein³

et al. 2023

Preprint

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We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, there is an algorithm that (1/2 − ε)-approximates the maximum path cover size of an n-vertextime algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give O(n) time algorithms that estimate the cost of graphic TSP and (1, 2)-TSP up to factors of 1.83 and (1.5 + ε), respectively. Our algorithm for graphic TSP improves over a 1.92-approximate O(n) time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. Our algorithm for (1, 2)-TSP improves over a folklore (1.75 + ε)-approximate O(n)-time algorithm, as well as a (1.625 + ε)-approximate O(n √ n)-time algorithm of [CHK ICALP'20].Our analysis of the running time uses connections to parallel algorithms and is informationtheoretically optimal up to poly log n factors. Additionally, we show that our approximation guarantees for path cover and (1, 2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

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Section: Introductionmentioning

confidence: 99%

Sublinear Algorithms for TSP via Path Covers

Behnezhad¹,

Roghani²,

Rubinstein³

et al. 2023

Preprint

View full text Add to dashboard Cite

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“…An equivalent view of the problem is that we are given a complete weighted undirected graph G(V, E) where the weights satisfy triangle inequality, and the goal is to find a Hamiltonian cycle of minimum weight. The study of metric TSP is intimately connected to many algorithmic developments, and the polynomial-time approximability of metric TSP and its many natural variants are a subject of extensive ongoing research (see, for instance, [3,13,18,20,22,[29][30][31][32][33] and references within for some relatively recent developments). In this paper, we consider the following question: can one design sublinear algorithms that can be used to obtain good estimates of the cost of an optimal TSP tour?…”

Section: Introductionmentioning

confidence: 99%

Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation

Chen,

Kannan,

Khanna

2020

Preprint

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We consider the problem of designing sublinear time algorithms for estimating the cost of minimum metric traveling salesman (TSP) tour. Specifically, given access to a n × n distance matrix D that specifies pairwise distances between n points, the goal is to estimate the TSP cost by performing only sublinear (in the size of D) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any ε > 0, there exists an Õ(n/ε O(1) ) time algorithm that returns a (1 + ε)-approximate estimate of the MST cost. This result immediately implies an Õ(n/ε O(1) ) time algorithm to estimate the TSP cost to within a (2 + ε) factor for any ε > 0. However, no o(n 2 ) time algorithms are known to approximate metric TSP to a factor that is strictly better than 2. On the other hand, there were also no known barriers that rule out existence of (1 + ε)-approximate estimation algorithms for metric TSP with Õ(n) time for any fixed ε > 0. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost.On the algorithmic side, we first consider the graphic TSP problem where the metric D corresponds to shortest path distances in a connected unweighted undirected graph. We show that there exists an Õ(n) time algorithm that estimates the cost of graphic TSP to within a factor of (2 − ε 0 ) for some ε 0 > 0. This is the first sublinear cost estimation algorithm for graphic TSP that achieves an approximation factor less than 2. We also consider another well-studied special case of metric TSP, namely, (1, 2)-TSP where all distances are either 1 or 2, and give an Õ(n 1.5 ) time algorithm to estimate optimal cost to within a factor of 1.625. Our estimation algorithms for graphic TSP as well as for (1, 2)-TSP naturally lend themselves to Õ(n) space streaming algorithms that give an 11/6-approximation for graphic TSP and a 1.625-approximation for (1, 2)-TSP. These results motivate the natural question if analogously to metric MST, for any ε > 0, (1 + ε)-approximate estimates can be obtained for graphic TSP and (1, 2)-TSP using Õ(n) queries. We answer this question in the negative -there exists an ε 0 > 0, such that any algorithm that estimates the cost of graphic TSP ((1, 2)-TSP) to within a (1 + ε 0 )-factor, necessarily requires Ω(n 2 ) queries. This lower bound result highlights a sharp separation between the metric MST and metric TSP problems.Similarly to many classical approximation algorithms for TSP, our sublinear time estimation algorithms utilize subroutines for estimating the size of a maximum matching in the underlying graph. We show that this is not merely an artifact of our approach, and that for any ε > 0, any algorithm that estimates the cost of graphic TSP or (1, 2)-TSP to within a (1 + ε)-factor, can also be used to estimate the size of a maximum matching in a bipartite graph to within an εn additive error. This connection allows us to translate known lower bounds for matching size estimation in various models to similar lower bounds ...

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Data Collection Task Planning of a Fixed-Wing Unmanned Aerial Vehicle in Forest Fire Monitoring

et al. 2021

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This article studies the data collection task planning for a fixed-wing unmanned aerial vehicle (UAV) in forest fire monitoring. Multiple wireless-based detection nodes (DNs) are distributed in highrisk areas of the forest to monitor the surrounding environment. The task of UAV is to circularly fly to them and collect the environmental data. Because of the kinematic constraints of UAV and the effective communication range between UAV and DN, this problem can be generally regarded as a Dubins traveling salesman problem with neighborhood (DTSPN). A bi-level hybridization-based metaheuristic algorithm (BLHMA) is proposed for solving this problem. At the first level, differential evolution (DE) optimizes the continuous-valued communication positions and UAV headings by the population-based search. For the asymmetric traveling salesman problem (ATSP) corresponding to the combination of the positions and headings generated by DE, a constructive heuristic based on self-organized multi-agent competition (SOMAC) is proposed to determine the discrete collection sequence. By competitive iterations in such a cooperative way in DE, a high-quality data collection tour can be generated. At the second level, a local search based on multistage approximate gradient is proposed to further refine the positions and headings, which accelerates the convergence of the BLHMA. Referring to a real-world scene of forest fire mornitoring, the simulation experiments are designed, and comparative results show that BLHMA can find significantly shorter data collection tours in most cases over three state-of-the-art algorithms. The proposed UAV data collection planning algorithm is conducive to the efficient execution of the forest fire monitoring data collection mission and the energy saving of UAV.

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Improved integrality gap upper bounds for traveling salesperson problems with distances one and two

Cited by 4 publications

References 22 publications

Sublinear Algorithms for TSP via Path Covers

Sublinear Algorithms for TSP via Path Covers

Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation

Data Collection Task Planning of a Fixed-Wing Unmanned Aerial Vehicle in Forest Fire Monitoring

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