C. H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science, 4:237-247, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Same as in NP1, except that the distance between two points is taken to be t v u w . NP3: Euclidean Traveling Salesman [GARE76b] Input: Same as in NP1. Output: A minimum length tour going through each point in X. The Euclidan distance measure is used. NP4: Euclidean Path Traveling Salesman [PAPA77] (also called Euclidean Hamiltonian Path) Input: Same is in NP1. Output: A minimum length path that visits all points in X exactly once. The Euclidean distance measure is used. NP5: Manhattan Traveling Salesman [GARE76b] Input: Same as in NP3. used. NP6: Manhattan Path Traveling Salesman ....

....(also called Euclidean Hamiltonian Path) Input: Same is in NP1. Output: A minimum length path that visits all points in X exactly once. The Euclidean distance measure is used. NP5: Manhattan Traveling Salesman [GARE76b] Input: Same as in NP3. used. NP6: Manhattan Path Traveling Salesman [PAPA77] (also called Manhattan Hamiltonian Path) Input: Same as in NP1. used. NP7: Chromatic Number I [EHRL76] Input: A graph G which is the intersection graph for straight line segments in the plane. Output: The minimum number of colors needed to color G. NP8: Chromatic Number II [EHRL76] Input: ....

Papadimitriou, C.H.",The Euclidean Traveling Salesman Problem is NP-Complete", Theoretical Computer Science 4, 1977, pp.237-244.

....instances, two key complexity questions have been answered. As follows from results of Itai, Papadimitriou, and Swarc ter [17] the Minimum TSP is NP hard for any xed dimension d and any L p or polyhedral norm; see also the earlier results by Garey, Graham, and Johnson [13] and Papadimitriou [23]. On the other hand, results of Arora [3] and Mitchell [21] imply that in all these cases a polynomial time approximation scheme (PTAS) exists, i.e. a sequence of polynomial time algorithms A k , 1 k 1, where A k is guaranteed to nd a tour whose length is within a ratio of 1 (1=k) of ....

Papadimitriou, C.H., \The Euclidean traveling salesman problem is NP-complete," Theoretical Comp. Sci. 4 (1977), 237-244.

....(respectively, k; s) formulas. That is, the (k; s) SAT problem is: given a (k; s) formula, is it satisfiable It is known that ( 3; 3) SAT is NP complete[1] Tovey[10] has shown that ( n; 2) SAT can be solved in polynomial time. When all clauses have the same number of variables, Papadimitriou[8] has shown that (3; 5) SAT is NP complete. Recently Tovey[10] has improved this to show that (3; 4) SAT is NP complete and that every (3; 3) formula is satisfiable. It follows easily that these NPcomplete results carry over to the MAXSAT problem (where we want to find the maximum number of clauses ....

C. H. Papadimitriou, "The Euclidean traveling salesman problem is NP-complete," Theoretical Computer Science 4 (1977) 237-244.

....algorithm is only possible for 203=202, unless P=NP [6] Because of these negative results, many people have attempted to solve special cases of this problem. Researchers have often exploited the di erent network topologies on which the TSP or more generally MVSP are formulated. Papadimitriou [14] shows that the TSP is NP complete even in the Euclidean 2 plane. For the general TSP, a 3=2 approximation is known, due to Christo des [4] Approximation algorithms are known for several special cases [8] Psaraftis et al. 15] consider SVSP on a path when all handling times are zero. They show ....

Papadimitriou, C. H. The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science 4, 3 (June 1977), 237-244.

....at Stony Brook, Stony Brook, NY 11794 3600. Partially supported by grants from HRL Laboratories, NASA Ames, the National Science Foundation (CCR9732220) Northrop Grumman Corporation, Sandia National Labs, and Sun Microsystems. gions (neighborhoods) are single points, and consequently is NP hard [6, 16]. Related Work. The TSP has a long and rich history of research in combinatorial optimization. It has been studied extensively in many forms, including geometric instances; see [3, 9, 10, 14, 18] The problem is known to be NP hard, even for points in the Euclidean plane [6, 16] It has recently ....

....is NP hard [6, 16] Related Work. The TSP has a long and rich history of research in combinatorial optimization. It has been studied extensively in many forms, including geometric instances; see [3, 9, 10, 14, 18] The problem is known to be NP hard, even for points in the Euclidean plane [6, 16]. It has recently been shown that the geometric instances of TSP (e.g. Euclidean TSP) have a polynomial time approximation scheme, as developed by Arora [2] and Mitchell [13] and later improved by Rao and Smith [17] Arkin and Hassin [1] were the rst to study approximation algorithms for the ....

C. H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoret. Comput. Sci., 4:237-244, 1977.

....TSP Introduction 1 The n city Euclidean Traveling Salesman Problem is the TSP where each city i is represented as a point p i =#x i ;y i #;x i ;y i 2R, in the plane and the distance c#p i ;p j # between any pair of cities i and j is computed according to the Euclidean metric, i; j =1; n. Papadimitriou #1977# proved the Euclidean TSP to be NP hard. We giveanO#mn# time and O#n# space algorithm for solving the special case of the n city Euclidean TSP where n #m cities lie on the boundary of the convex hull of the n cities, and the other m cities lie on a line segment inside this convex hull. This ....

Papadimitriou, C. H. #1977#, The Euclidean traveling salesman problem is NP - complete, Theoretical Computer Science 4, 237#244.

....1 The n city Euclidean Traveling Salesman Problem is the TSP where each city i is represented as a point p i = x i ; y i ) x i ; y i 2 R, in the plane and the distance c(p i ; p j ) between any pair of cities i and j is computed according to the Euclidean metric, i; j = 1; n. Papadimitriou [1977] proved the Euclidean TSP to be NP hard. We give an O(mn) time and O(n) space algorithm for solving the special case of the n city Euclidean TSP where n Gamma m cities lie on the boundary of the convex hull of the n cities, and the other m cities lie on a line segment inside this convex hull. ....

Papadimitriou, C. H. (1977), The Euclidean traveling salesman problem is NP - complete, Theoretical Computer Science 4, 237--244.

....the TSP where the cities are represented by points in the two dimensional Euclidean plane and the distances are measured according to the Euclidean metric. We will write d(x; y) to denote the Euclidean distance between points x and y. This special TSP case is still NP hard (see e.g. Papadimitriou [83] or Chapter 3 in [71] However, in the Euclidean case the shortest TSP tour does not intersect itself and thus geometry makes the problem somewhat easier. Section 3.1 deals with some simple special cases that result from convex sets. Section 3.2 deals with the case where the cities lie on a small ....

C.H. Papadimitriou, The Euclidean traveling salesman problem is NP-complete, Theoretical Computer Science 4, 1977, 237--244.

....of the buyers neighborhoods and finally returns to his initial departure point. Note that the neighborhoods may overlap partially. The problem generalizes the Euclidean Traveling Salesman Problem (TSP) in which the areas specified by the buyers are single points, and consequently it is NP hard [Pa, GGJ]. On the other hand, it is known that the optimal tour of a Euclidean Traveling Salesman (and in fact any symmetric TSP obeying the triangle inequality) can be approximated by a tour of length at most one and a half times the optimal tour [Ch] Such approximation algorithms are available also for ....

Papadimitriou, C.H. (1977), "The Euclidean Traveling Salesman Problem is NPComplete ", Theor. Comput. Sci., 4, 237-244.

....Each edge of such a tour has a length that is equal to the Euclidean distance between its endpoints. The length of a tour is the sum of the lengths of all its edges. The TSP is to compute a tour along the points of S of minimal length. Since this problem is NP complete for dimension d 2 (see [6]) it is natural to consider the weaker problem of designing efficient algorithms that approximate the optimal tour. We call a tour having length at most r times the length of an optimal tour an r approximate TSP tour. It is well known that for d = 2, a 2 approximate TSP tour can be computed in ....

C.H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science 4 (1977), pp. 237-244.

....is a variant of the problem in which one wants a shortest path that visits S. The TSP is a classical problem in combinatorial optimization, and has been studied extensively in many forms, including geometric instances; see [59, 250, 343, 230] The problem is NP hard, as shown by Papadimitriou [316], even for points in the Euclidean plane. The TSP has a simple approximation algorithm based on doubling the minimum spanning tree. Since an optimal tour spans all of the sites (and is converted into a spanning tree by deleting any one edge) it must be at least as long as the minimum spanning ....

C. H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoret. Comput. Sci., 4:237--244, 1977.

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C. H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science, 4:237-247, 1977.

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