Improved Lower Bounds on the Approximability of the Traveling Salesman Problem

Böckenhauer, Hans-Joachim; Seibert, Sebastian

doi:10.1051/ita:2000115

Cited by 33 publications

(24 citation statements)

References 12 publications

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“…The best known standard-approximation ratio known for min TSP12 is 7/6 ([23]), while the best known standard inapproximability bound is 743/742 − , for any > 0 ( [16]). The APX-hardness of ∆ α STSP and ∆ α RTSP is proved in [10] and [6], respectively; the precise inapproximability bounds are (7612 + 8α 2 + 4α)/(7611 + 10α 2 + 5α) − ∀ > 0, for the former, and (3804 + 8α)/(3803 + 10α) − ∀ > 0, for the latter. In the opposite, max TSP is approximable within standard-approximation ratio 3/4 ( [25]).…”

Section: Definitionmentioning

confidence: 99%

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Approximation algorithms for the traveling salesman problem

Monnot

Paschos

Toulouse

2003

Mathematical Methods of Operations Research (ZOR)

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We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst-and the best-value solutions of an instance. We next show that the 2 OPT, one of the mostknown traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2 OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, for any > 0, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 + and traveling salesman with distances 1 and 2 within better than 741/742 + . Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now.

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Section: Definitionmentioning

confidence: 99%

“…A special but very natural case of TSP, commonly called metric TSP and denoted by ∆TSP in what follows, is the one where edge-distances satisfy the triangle inequality. Recently, researchers are interested in metric min TSP-instances defined on parameterized 1 triangle inequalities ( [2,6,9,8,7,10]). Consider α ∈ Q.…”

Section: Introductionmentioning

confidence: 99%

Approximation algorithms for the traveling salesman problem

Monnot

Paschos

Toulouse

2003

Mathematical Methods of Operations Research (ZOR)

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“…For any β > 1 2 , we have a constant polynomial-time approximation ratio, depending on β only. Böckenhauer and Seibert [8] proved that ∆ β -TSP is APX-hard for every β > 1 2 (note that for β = 1 2 , the problem becomes trivially solvable in polynomial time). Here, we prove that lm-∆ β -TSP is NP-hard for every β > 1 2 .…”

Section: Introductionmentioning

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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“…[BHK + 00] designed three algorithms for TSP subproblems with instances satisfying the α-strengthen triangle inequality, which yield the approximation ratios starting with 1 for α = 1/2 and growing with α to 3/2 for α = 1. A very strong result has been proved by Böckenhauer and Seibert [BS00] who established an explicit lower bound on polynomial time approximability of TSP with sharped triangle inequality for any α > 1/2 and this lower bounds grows with α. Thus, the TSP instances with weights from the interval [1, 1 + ε] form an APX-hard problem for arbitrary small ε > 0.…”

Section: Conclusion and An Overviewmentioning

confidence: 97%

Stability of Approximation in Discrete Optimization

Hromkovič

IFIP International Federation for Information Processing

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One can try to parametrize the set of the instances of an optimization problem and look for in polynomial time achievable approximation ratio with respect to this parametrization. When the approximation ratio grows with the parameter, but is independent of the size of the instances, then we speak about stable approximation algorithms. An interesting point is that there exist stable approximation algorithms for problems like TSP that is not approximable within any polynomial approximation ratio in polynomial time (assuming P is not equal to NP). The investigation of the stability of approximation overcomes in this way the troubles with measuring the complexity and approximation ratio in the worst-case manner, because it may success in partitioning of the set of all input instances of a hard problem into infinite many classes with respect to the hardest of the particular inputs. We believe that approaches like this will become the core of the algorithmics, because they provide a deeper insight in the hardness of specific problems and in many application we are not interested in the worstcase problem hardness, but in the hardness of forthcoming problem instances.

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Improved Lower Bounds on the Approximability of the Traveling Salesman Problem

Cited by 33 publications

References 12 publications

Approximation algorithms for the traveling salesman problem

Approximation algorithms for the traveling salesman problem

Reusing Optimal TSP Solutions for Locally Modified Input Instances

Stability of Approximation in Discrete Optimization

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