Reoptimization of Traveling Salesperson Problems: Changing Single Edge-Weights

Berg, Tobias; Hempel, Harald

doi:10.1007/978-3-642-00982-2_12

Cited by 15 publications

(10 citation statements)

References 3 publications

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“…To this end, the authors developed their study by classifying the approximability of TSP depending on the metricity of both the input and the modified instance. More formally, they proved that: (i) if the input and the modified instance obey the strengthened triangle inequality, the problem admits a PTAS (this compares favorably with the APX-hardness of the counterpart); (ii) if the input and the modified instance are both metric, the problem can be approximated within a factor of 7/5, which compares favorably with the approximation ratio guaranteed by the Christofides algorithm; this result was further improved in [6], where the authors proved the problem can be approximated within a factor of 4/3; and finally, (iii) if the input and the modified instance obey the relaxed triangle inequality with 1 < β < 3.34899, the problem admits a β 2 15β 2 +5β−6 13β 2 +3β−6 -approximation algorithm, which is better than its counterparts given in [5,17]. In other words, for the considered range of values of β, all these reapproximation algorithms are stable.…”

Section: The Reoptimization Version Of the Tspmentioning

confidence: 97%

Stability of Reapproximation Algorithms for the $$\beta $$-Metric Traveling Salesman (Path) Problem

D’Andrea

Forlizzi

Proietti

2018

Adventures Between Lower Bounds and Higher Altitudes

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Inspired by the concept of stability of approximation, we consider the following (re)optimization problem: Given a minimum-cost Hamiltonian cycle of a complete non-negatively real weighted graph G = (V, E, c) obeying the strengthened triangle inequality (i.e., for some strength factor 1/2 ≤ β < 1, we have that ∀u, v, z ∈ V, c(u, z) ≤ β(c(u, v) + c(v, z))), and given a vertex v whose removal from G (resp., addition to G), along with all its incident edges, produces a new weighted graph still obeying the strengthened triangle inequality, find a minimumcost Hamiltonian cycle of the modified graph. This problem is known to be NP-hard, but we show that it admits a PTAS, which just consists of either returning the old optimal cycle (after having bypassed the removed node), or instead computing (for finitely many inputs) a new optimal solution from scratch − depending on the required accuracy in the approximation. Then, we turn our attention to the case in which a minimum-cost Hamiltonian path is given instead, and the underlying graph obeys the relaxed triangle inequality. Here, if one edge weight is increased, and βO, βM ≥ 1 denotes the relaxation factor of the original and the modified graph, respectively, then we show how to obtain an approximation of 1 + 2 min{βO, βM}, which improves over existing solutions as soon as min{βO, βM} ≥ 3+2 √ 6 5

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Section: The Reoptimization Version Of the Tspmentioning

confidence: 97%

Stability of Reapproximation Algorithms for the $$\beta $$-Metric Traveling Salesman (Path) Problem

D’Andrea

Forlizzi

Proietti

2018

Adventures Between Lower Bounds and Higher Altitudes

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“…The concept of reoptimization was first mentioned by Schäffter [31] in the context of postoptimality analysis for a scheduling problem. Since then, the concept of reoptimization has been investigated for several different problems, including the traveling salesman problem [1,5,7,15,16,30], the minimum latency problem [4,26], the rural postman problem [3], fast reoptimization of the spanning tree problem [23], the knapsack problem [2], covering problems [12], the shortest common superstring problem [10], maximum-weight induced hereditary problems [22], and scheduling [6,21,31]. There are several overviews on reoptimization [4,19,24,33].…”

Section: Related Workmentioning

confidence: 99%

Robust Reoptimization of Steiner Trees

Goyal¹,

Mömke

2020

Algorithmica

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In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed > 0 , approximating the reoptimization problem with respect to a given (1 + )-approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

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“…The following facts on the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems [10] are well known. Under the assumption that P ¹ NP, there exist no FPTAS for the traveling salesman tour minimization problem with triangle inequality with increasing (decreasing) the weight of a unique edge.…”

Section: Definitionmentioning

confidence: 99%

On the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems

Mikhailyuk

2011

Cybern Syst Anal

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It is shown that no polynomial-time approximation scheme exists for the reoptimization of the set covering problem in inserting an element into or eliminating it from any set. A similar result is obtained for the minimum graph coloring problem in inserting a vertex with at most two incidence edges and for the minimal bin packing problem in eliminating any element.The concept of reoptimization [1-7] is as follows. Let Ï be some NP-hard (or, possibly, NP-complete) problem, and let I be the initial instance of the problem Ï whose optimal solution is known. A new instance I¢ of the problem Ï is proposed that is obtained as a result of some "insignificant" changes in the instance I. The following question arises: how can knowledge on an optimal solution to I be efficiently used for the computation of an exact or approximate solution to the instance I¢? In using approximate methods, the objective of reoptimization is the application of knowledge on the solution of the initial instance I with a view to achieving a better approximation quality (approximation ratio) of I¢, or creating a more (time) efficient algorithm for the determination of an optimal (or close to the optimal) solution of I¢, or attaining both goals.Investigations on the reoptimization of various discrete optimization problems are performed in [1][2][3][4][5][6][7]. PRELIMINARY INFORMATION ON APPROXIMATE ALGORITHMS AND APPROXIMATION CLASSES OF DISCRETE OPTIMIZATION PROBLEMSWe give some concepts necessary for further presentation and introduced in [7]. Definition 1. An NP-optimization (NPO) problem Ï is defined as a quadruple ( , , , ) E m Sol opt such that · E is the set of instances of Ï recognized in polynomial time; · for a given I E Î , Sol( ) I is the set of feasible solutions to I; for each S I S Î Sol( ), | | (the size of S ) is polynomial with respect to | | I (the size of I); for any given I and any S (polynomial with respect to | | I ), S I Î Sol( ) can be determined in polynomial time;· for given I E Î and S I Î Sol( ), m I S ( , ) is the (numerical) value of S and m is a polynomially computable quantity called an objective function;· opt {min } Î , max is the type of an optimization problem. For an NPO problem Ï { Sol opt} = E m , , , , we denote an optimal solution to an instance I of Ï by S I * ( ) and its value m I S I ( , ( ) ) * by opt ( ) I . Definition 2.For an NPO problem Ï Sol opt = ( , , , ) E m , an approximate (approximation) algorithm A is an algorithm that, for a given instance I of Ï produces a feasible solution S I Î Sol( ) .If A is executed in polynomial time with respect to | | I , then A is called a polynomial-time approximate algorithm for Ï . 368

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Reoptimization of Traveling Salesperson Problems: Changing Single Edge-Weights

Cited by 15 publications

References 3 publications

Stability of Reapproximation Algorithms for the $$\beta $$-Metric Traveling Salesman (Path) Problem

Stability of Reapproximation Algorithms for the $$\beta $$-Metric Traveling Salesman (Path) Problem

Robust Reoptimization of Steiner Trees

On the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems

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