Complexity and approximability of the Euclidean generalized traveling salesman problem in grid clusters

We consider the geometric version of the well-known Generalized Traveling Salesman Problem introduced in 2015 by Bhattacharya et al. that is called the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC). They proved the intractability of the problem and proposed first polyn...

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Vydané v:Annals of mathematics and artificial intelligence Ročník 88; číslo 1-3; s. 53 - 69
Hlavní autori: Khachay, Michael, Neznakhina, Katherine
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Cham Springer International Publishing 01.03.2020
Springer
Springer Nature B.V
Predmet:
Algorithms
Approximation
Artificial Intelligence
Clusters
Complex Systems
Computer Science
Euclidean geometry
Management science
Mathematical analysis
Mathematics
Polynomials
Traveling salesman problem
Generalized traveling salesman problem
Polynomial time approximation scheme
90C27
Polynomial time solvable subclass
ISSN:1012-2443, 1573-7470
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Shrnutí:We consider the geometric version of the well-known Generalized Traveling Salesman Problem introduced in 2015 by Bhattacharya et al. that is called the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC). They proved the intractability of the problem and proposed first polynomial time algorithms with fixed approximation factors. The extension of these results in the field of constructing the polynomial time approximation schemes (PTAS) and the description of non-trivial polynomial time solvable subclasses for the EGTSP-GC appear to be relevant in the light of the classic C. Papadimitriou result on the intractability of the Euclidean TSP and recent inapproximability results for the Traveling Salesman Problem with Neighborhoods (TSPN) in the case of discrete neighborhoods. In this paper, we propose Efficient Polynomial Time Approximation Schemes (EPTAS) for two special cases of the EGTSP-GC, when the number of clusters k = O ( log n ) and k = n − O ( log n ) . Also, we show that any time, when one of the grid dimensions (height or width) is fixed, the EGTSP-GC can be solved to optimality in polynomial time. As a consequence, we specify a novel non-trivial polynomially solvable subclass of the Euclidean TSP in the plane.
Bibliografia:ObjectType-Article-1
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ISSN:1012-2443
1573-7470
DOI:10.1007/s10472-019-09626-w