On the Stability of Approximation for Hamiltonian Path Problems

Forlizzi, Luca; Hromkovič, Juraj; Proietti, Guido; Seibert, Sebastian

doi:10.1007/978-3-540-30577-4_18

Cited by 6 publications

(11 citation statements)

References 8 publications

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“…Then, we turn our attention to a more general version of the TSP, called the Traveling Salesman Path Problem (TSPP for short), in which we are also given two vertices s, t ∈ V and the goal is to find a path from s to t visiting each vertex in V exactly once. More precisely, we address the relaxed metric TSPP (Δ β>1 -TSPP for short), whose input instances obey the relaxed triangle inequality, and whose goal is the same as for the TSPP, i.e., finding a minimum-cost Hamiltonian path between the given vertices s and t. The best current approximation algorithm for this problem is the so-called Path Matching Christofides Algorithm-TSPP [25], with an approximation ratio of 5 3 β 2 . Also in this case, we focus on the reoptimization version of the relaxed metric TSPP.…”

Section: Our Resultsmentioning

confidence: 99%

“…Then the parameter β can be used to partition the set of all input instances of a hard optimization problem (from β = 1/2, i.e., unweighted graphs, up to β = ∞, i.e., general weighted graphs), and to classify them according to their respective computational properties. This idea proved to be very fruitful, giving rise to numerous results for different hard optimization problems, such as k-connectivity [11,12], Steiner tree [8,9,14,19], Traveling Salesman Problem (TSP for short) [1,2,5,[15][16][17]22], and its variants [7,10,13,18,20,21,24,25].…”

Section: Introductionmentioning

confidence: 99%

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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: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…Win/win strategies fit well into the framework of parameterized complexity [11,18] as well as stability of approximation [5,14,7], because all of these approaches are based on studying the "hardness" of their problem instances.…”

Section: Related Known Resultsmentioning

confidence: 99%

Structural Properties of Hard Metric TSP Inputs

Mömke

2011

SOFSEM 2011: Theory and Practice of Computer Science

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The metric traveling salesman problem is one of the most prominent APX-complete optimization problems. An important particularity of this problem is that there is a large gap between the known upper bound and lower bound on the approximability (assuming P = N P ). In fact, despite more than 30 years of research, no one could find a better approximation algorithm than the 1.5-approximation provided by Christofides. The situation is similar for a related problem, the metric Hamiltonian path problem, where the start and the end of the path are prespecified: the best approximation ratio up to date is 5/3 by an algorithm presented by Hoogeveen almost 20 years ago.In this paper, we provide a tight analysis of the combined outcome of both algorithms. This analysis reveals that the sets of the hardest input instances of both problems are disjoint in the sense that any input is guaranteed to allow at least one of the two algorithms to achieve a significantly improved approximation ratio. In particular, we show that any input instance that leads to a 5/3-approximation with Hoogeveen's algorithm enables us to find an optimal solution for the traveling salesman problem. This way, we determine a set S of possible pairs of approximation ratios. Furthermore, for any input we can identify one pair of approximation ratios within S that forms an upper bound on the achieved approximation ratios.

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“…More precisely, in order to find a Hamiltonian path between a given pair of vertices in a β-metric graph, we will employ the algorithm by Forlizzi et al [11], a variation of the path-matching Christofides algorithm (PMCA, see [5]) for the path version of near-metric TSP, which yields an approximation guarantee of …”

Section: The Near-metric Casementioning

confidence: 99%

“…2. For all {f, f } ∈ E, compute a Hamiltonian path between the two vertices from (f ∪ f ) \ e on the graph G \ (e ∩ (f ∪ f )), using the PMCA path variant by Forlizzi et al [11]. Augment this path by edges f , f , and, if cO(e) > cN (e), edge e to obtain the cycle C {f,f } .…”

Section: Algorithmmentioning

confidence: 99%

Reusing Optimal TSP Solutions for Locally Modified Input Instances

Böckenhauer

Forlizzi

Hromkovič

et al.

Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006

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Abstract. Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U ) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U . Here, we give non-trivial examples both of problems where this is and problems where this is not the case. Our main results are these:1. The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem lm-TSP which is as hard as TSP itself, i. e., unless P = N P , there is no polynomial-time p(n)-approximation algorithm for lm-TSP for any polynomial p. Moreover, lm-TSP where inputs must satisfy the β-triangle inequality (lm-∆ β -TSP) remains NP-hard for all β > 1 2 . 2. For lm-∆-TSP (i. e., metric lm-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. 3. Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for lm-∆ β -TSP than for ∆ β -TSP. 4. Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem.

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On the Stability of Approximation for Hamiltonian Path Problems

Cited by 6 publications

References 8 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

Structural Properties of Hard Metric TSP Inputs

Reusing Optimal TSP Solutions for Locally Modified Input Instances

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