Cutler, M. #1980#, E#cient special case algorithms for the N-line planar traveling salesman problem, Networks 10, 183#195. Home/Search Document Not in Database Summary Related Articles Check |
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The Convex-Hull-and-Line Traveling Salesman Problem: A.. - Deineko, Dal, Rote (1992) (Correct)
....lie on one line, an optimal tour has the property that the cities on the boundary of the convex hull of the cities are visited in their cyclic order. Note that the case where all cities lie on two parallel lines corresponds to the case where all cities lie on the boundary of their convex hull. Cutler #1980# has given an O#n 3 # time and O#n 2 # space dynamic programming algorithm for solving the so called 3 line TSP, i.e. the Euclidean TSP where all points lie on three distinct parallel lines in the plane. Rote #1992# extended the results of Cutler by considering the N line TSP, i.e. the ....
....chain #u 1 ;g i ;g j ;l 1 #or#u 1 ;g j ;g i ;l 1 #, thereby creating an intersection. Similarly,ifj#mthen the segment cannot be inserted between u p and l q without introducing an intersection. Remark. The above lemma corresponds to Cutler s Triangle Theorem with respect to the 3 line TSP; see Cutler #1980#. The Triangle Theorem states that at least one of the edges fu 1 ;l 1 g; fu 1 ;g 1 gand fl 1 ;g 1 g is an edge of an optimal tour. Similarly,at least one of the edges fu p ;l q g; fu p ;g mg and fl q ;g mg is an edge of an optimal tour. Therefore, the cases mentioned in the above lemmahave to ....
Cutler, M. #1980#, E#cient special case algorithms for the N-line planar traveling salesman problem, Networks 10, 183#195.
The N-line Traveling Salesman Problem - Rote (1991) (Correct)
....June 1991 Abstract The special case of the Euclidean Traveling Salesman Problem, where the n given points lie on a small number #N#of parallel lines in the plane, is solved by a dynamic programming approach in time n N , for #xed N, i.e. in polynomial time. This extends a result of Cutler #1980# for 3 lines. Such problems arise for example in the fabrication of printed circuit boards, where the distance traveled by a laser which drills holes in certain places of the board should be minimized. The parallelity condition can be relaxed to point sets which lie on N #almost parallel line ....
....Such problems arise in practical applications, for example, in the manufacturing of printed circuit boards and related devices. Nevertheless, since the high running times are rather high, the algorithm seems to be of theoretical interest only. Our algorithm is an extension of an algorithm by Cutler #1980# for three parallel lines. #For two lines, the problem is trivial.# Cutler also considered the Traveling Salesman Path Problem. A similar but easier special case was considered by Ratli# and Rosenthal #1983#: the problem of order picking in a rectangular warehouse. These authors also used the ....
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M. Cutler #1980# E#cient special case algorithms for the N-line planar Traveling Salesman Problem, Networks 10, 183#195.
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