Improved approximation algorithms for some min-max and minimum cycle cover problems

Yu, Wei; Liu, Zhaohui

doi:10.1016/j.tcs.2016.01.041

Cited by 34 publications

(13 citation statements)

References 15 publications

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“…We propose an approximation algorithm that runs in polynomial time with a fixed number of depots. The approximation ratio of 5 + is comparable to the state-of-the-art algorithm for unrooted problem [7]. Moreover, even though our formulation is somewhat different, our approximation ratio is better than the previous best algorithm for rooted cycle cover problem [7].…”

Section: Related Workmentioning

confidence: 76%

“…Similarly, in an independent study, Xu et al [6] investigated the same cycle cover problems and proposed algorithms with approximation ratio of 5 1 3 + and 6 1 3 + for unrooted and (uncapacitated) rooted min-max cycle cover problems, respectively. 1 In addition, Yu and Liu [7] proposed algorithms with improved approximation ratio of 5 + and 6 + for unrooted and rooted min-max cycle cover problems, respectively, by utilizing the well-known Christofides algorithm [8] for the TSP problem.…”

Section: Related Workmentioning

confidence: 99%

“…This result is generalized by Xu et al [15] who showed that there does not exist a polynomial time algorithm for the unrooted and rooted min-max cycle cover problems with an approximation ratio less than 1 1 3 unless P = N P . It is worth mentioning that, although our rooted min-max cycle cover formulation is inspired by [6] and [7], our model is slightly different: they only required the union of cycles to cover all vertices, except for depots. In other words, the depots need not be covered by the cycle cover.…”

Section: Related Workmentioning

confidence: 99%

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Miniature Robot Path Planning for Bridge Inspection: Min-Max Cycle Cover-Based Approach

Lin

2020

Preprint

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We study the problem of planning the deployments of a group of mobile robots. While the problem and formulation can be used for many different problems, here we use a bridge inspection as the motivating application for the purpose of exposition. The robots are initially stationed at a set of depots placed throughout the bridge. Each robot is then assigned a set of sites on the bridge to inspect and, upon completion, must return to the same depot where it is stored.The problem of robot planning is formulated as a rooted minmax cycle cover problem, in which the vertex set consists of the sites to be inspected and robot depots, and the weight of an edge captures either (i) the amount of time needed to travel from one end vertex to the other vertex or (ii) the necessary energy expenditure for the travel. In the first case, the objective function is the total inspection time, whereas in the latter case, it is the maximum energy expenditure among all deployed robots. We propose a novel algorithm with approximation ratio of 5 + , where 0 < < 1. In addition, the computational complexity of the proposed algorithm is shown to be O n 2 +2 m−1 n log(n+k) , where n is the number of vertices, and m is the number of depots.

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Section: Related Workmentioning

confidence: 76%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Miniature Robot Path Planning for Bridge Inspection: Min-Max Cycle Cover-Based Approach

Lin

2020

Preprint

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“…The first four lines of the algorithm partition the vertices according to their latency constraints. For each of those partitions, the function MCCP(V, λ) called in line 6 uses an approximation algorithm for the minimum cycle cover problem from [25]. Then, the appropriate number of robots are placed on each cycle returned by the MCCP function to satisfy the latency constraints.…”

Section: A O(log ρ) Approximationmentioning

confidence: 99%

Multi-Robot Routing for Persistent Monitoring with Latency Constraints

Asghar

Smith

Sundaram

2019

2019 American Control Conference (ACC)

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In this paper we study a multi-robot path planning problem for persistent monitoring of an environment. We represent the areas to be monitored as the vertices of a weighted graph. For each vertex, there is a constraint on the maximum time spent by the robots between visits to that vertex, called the latency, and the objective is to find the minimum number of robots that can satisfy these latency constraints. The decision version of this problem is known to be PSPACE-complete. We present a O(log ρ) approximation algorithm for the problem where ρ is the ratio of the maximum and the minimum latency constraints. We also present an orienteering based heuristic to solve the problem and show through simulations that in most of the cases the heuristic algorithm gives better solutions than the approximation algorithm. We evaluate our algorithms on large problem instances in a patrolling scenario and in a persistent scene reconstruction application. We also compare the algorithms with an existing solver on benchmark instances.

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“…. , C k with (C i ) ≤ 1 for all i with minimum k. Recently, Yu et al [23,25] provided new approximation algorithms for the these problems with approximation ratios of 5 and 4 + 4/7 and running times of O(n 3 ) and O(n 5 ) respectively. Here and in the following n := |V |.…”

Section: Introductionmentioning

confidence: 99%

A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

Traub

Tröbst

2021

Math. Program.

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We consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.

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Improved approximation algorithms for some min-max and minimum cycle cover problems

Cited by 34 publications

References 15 publications

Miniature Robot Path Planning for Bridge Inspection: Min-Max Cycle Cover-Based Approach

Miniature Robot Path Planning for Bridge Inspection: Min-Max Cycle Cover-Based Approach

Multi-Robot Routing for Persistent Monitoring with Latency Constraints

A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

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