On the worst‐case performance of some algorithms for the asymmetric traveling salesman problem

Frieze, Alan; Galbiati, Giulia; Maffioli, Francesco

doi:10.1002/net.3230120103

Cited by 216 publications

(164 citation statements)

References 17 publications

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“…Since pref(s i ; s j ) jsj, we can conclude that opt(TSP) opt(S), where opt(TSP) is the optimal solution to TSP de ned above. This TSP is directed (sometimes called asymmetric); thus the best known approximation [9] is only within a factor of O(log n). Therefore, we must exploit more of the structure of the problem in order to achieve better bounds.…”

Section: Preliminariesmentioning

confidence: 99%

A 2 2/3-approximation algorithm for the shortest superstring problem

Armen

Stein

1996

Combinatorial Pattern Matching

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Given a collection of strings S = fs 1 ; : : :; s n g over an alphabet , a superstring of S is a string containing each s i as a substring; that is, for each i, 1 i n, contains a block of js i j consecutive characters that match s i exactly. The shortest superstring problem is the problem of nding a superstring of minimum length. The shortest superstring problem has applications in both data compression and computational biology. In data compression, the problem is a part of a general model of string compression proposed by Gallant, Maier and Storer (JCSS '80). Much of the recent interest in the problem is due to its application to DNA sequence assembly.The problem has been shown to be NP-hard; in fact, it was shown by Blum et al.(JACM '94) to be MAX SNP-hard. The rst O(1)-approximation was also due to Blum et al., who gave an algorithm that always returns a superstring no more than 3 times the length of an optimal solution. Several researchers have published results that improve on the approximation ratio; of these, the best previous result is our algorithm ShortString, which achieves a 2 3 4 {approximation (WADS '95).We present our new algorithm, G-ShortString, which achieves a ratio of 2 2 3 . It generalizes the ShortString algorithm, but the analysis di ers substantially from that of ShortString. Our previous work identi ed classes of strings that have a nested periodic structure, and which must be present in the worst case for our algorithms. We introduced machinery to descibe these strings and proved strong structural properties about them. In this paper we extend this study to strings that exhibit a more relaxed form of the same structure, and we use this understanding to obtain our improved result.

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Section: Preliminariesmentioning

confidence: 99%

A 2 2/3-approximation algorithm for the shortest superstring problem

Armen

Stein

1996

Combinatorial Pattern Matching

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“…Frieze, Galbiati and Maffioli [7] designed an approximation algorithm for ATSP with approximation ratio O(log n). Blaser [3] notes that the approximation ratio proved in [7] is precisely log 2 n (with leading constant 1), and then designs an algorithm for which he shows an approximation ratio of 0.999 log 2 n. Subsequently, Kaplan et al [10] designed an algorithm with approximation ratio 4/3 log 3 n 0.842 log 2 n (using a technique that they apply to other related problems as well).…”

Section: Introductionmentioning

confidence: 99%

“…Blaser [3] notes that the approximation ratio proved in [7] is precisely log 2 n (with leading constant 1), and then designs an algorithm for which he shows an approximation ratio of 0.999 log 2 n. Subsequently, Kaplan et al [10] designed an algorithm with approximation ratio 4/3 log 3 n 0.842 log 2 n (using a technique that they apply to other related problems as well). In this paper, we provide a modest improvement in the leading constant of the approximation ratio.…”

Section: Introductionmentioning

confidence: 99%

Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs

Feige

Singh

2007

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. In metric asymmetric traveling salesperson problems the input is a complete directed graph in which edge weights satisfy the triangle inequality, and one is required to find a minimum weight walk that visits all vertices. In the asymmetric traveling salesperson problem (ATSP) the walk is required to be cyclic. In asymmetric traveling salesperson path problem (ATSPP), the walk is required to start at vertex s and to end at vertex t.We improve the approximation ratio for ATSP from 4 3 log 3 n 0.84 log 2 n to 2 3 log 2 n. This improvement is based on a modification of the algorithm of Kaplan et al [JACM 05] that achieved the previous best approximation ratio. We also show a reduction from ATSPP to ATSP that loses a factor of at most 2 + in the approximation ratio, where > 0 can be chosen to be arbitrarily small, and the running time of the reduction is polynomial for every fixed . Combined with our improved approximation ratio for ATSP, this establishes an approximation ratio of ( 4 3 + ) log 2 n for ATSPP, improving over the previous best ratio of 4 log e n 2.76 log 2 n of Chekuri and Pal [Approx 2006].

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“…It uses different principles of soccer to solve combinatorial optimization problems. The quality of this technique is demonstrated applying it to four combinatorial problems [23]: Asymmetric traveling salesman problem (ATSP) [24], Vehicle Routing Problem with Backhauls (VRPB) [25], [26], n-Queen Problem (NQP) [27], One-Dimensional Bin Packing Problem (BPP) [28].This algorithm is a promising metaheuristic to solve combinatorial optimization problems [23].…”

Section: A Golden Ball Metaheuristicmentioning

confidence: 99%

Hybrid Algorithm for Solving the Quadratic Assignment Problem

Riffi¹,

Sayoti²

2019

IJIMAI

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T HE quadratic assignment problem (QAP) is one of the known classical combinatorial optimization problems, in 1976 Sahni and Gonzalez [1] proved that the QAP belongs to the class of NP-hard problems [1]. It was introduced for the first time by Koopmans and Beckmann in 1957 [2]; its purpose is to assign n facilities to n fixed locations with a given flow matrix of facilities and distance matrix of locations in order to minimize the total assignment cost. This problem is applied in various fields such as hospital layout [3], scheduling parallel production lines [4] and analyzing chemical reactions for organic compounds [5].Many recent hybrid approaches have improved performance in solving QAP such as genetic algorithm hybridized with tabu search method [6], ant colony optimization mixed with local search method [7] and ant colony optimization combined with genetic algorithm and local search method [8]. Recently the hybrid algorithms are much proposed and used by many researchers to find optimal or near optimal solutions for the QAP.In this paper we propose a new competitive approach when compared with other existing methods in the literature. The golden ball algorithm mixed with simulated annealing (GBSA) is considered here as a hybrid metaheuristic to apply in the quadratic assignment problem.This work presents an efficient adaptation of GBSA algorithm to the quadratic assignment problem (QAP). This algorithm is based on the concept of soccer; it guides the search by simulated annealing [9] to escape from the local optima. The suggested technique has never been proposed or tested with QAP. In this research we use some small, medium and large test problems for comparing our approach to other recent methods from literature. Our approach is able to explore effectively the search space; it reaches the known optimal solutions in less time.The rest of this paper is structured as follows: In section I, Introduction. In section II, Quadratic assignment problem formulation. In section III, Methods. In section IV, Results and discussion. In section V, Conclusion. KeywordsCombinatorial Optimization, Golden Ball Algorithm, Simulated Annealing, Quadratic Assignment Problem. AbstractThe Quadratic Assignment Problem (QAP) is a combinatorial optimization problem; it belongs to the class of NP-hard problems. This problem is applied in various fields such as hospital layout, scheduling parallel production lines and analyzing chemical reactions for organic compounds. In this paper we propose an application of Golden Ball algorithm mixed with Simulated Annealing (GBSA) to solve QAP. This algorithm is based on different concepts of football. The simulated annealing search can be blocked in a local optimum due to the unacceptable movements; our proposed strategy guides the simulated annealing search to escape from the local optima and to explore in an efficient way the search space. To validate the proposed approach, numerous simulations were conducted on 64 instances of QAPLIB to compare GBSA with existing algorithms in the literatu...

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On the worst‐case performance of some algorithms for the asymmetric traveling salesman problem

Cited by 216 publications

References 17 publications

A 2 2/3-approximation algorithm for the shortest superstring problem

A 2 2/3-approximation algorithm for the shortest superstring problem

Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs

Hybrid Algorithm for Solving the Quadratic Assignment Problem

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