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Showing papers on "Longest path problem published in 1974"


Proceedings ArticleDOI
30 Apr 1974
TL;DR: This paper shows that a number of NP-complete problems remain NP- complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP-complete.
Abstract: It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of SIMPLE MAX CUT (MAX CUT with edge weights restricted to value 1), and, as a corollary, the completeness of the OPTIMAL LINEAR ARRANGEMENT problem. We then show that even if the domains of the NODE COVER and DIRECTED HAMILTONIAN PATH problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For GRAPH 3-COLORABILITY, NODE COVER, and UNDIRECTED HAMILTONIAN CIRCUIT, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.

648 citations


Journal ArticleDOI
Frank Rubin1
TL;DR: It is shown that the original claim of generality for the path cost function is incorrect, and a restriction, called the pathconsistency property, is introduced, and the Lee algorithm holds for those path cost functions having this property.
Abstract: The Lee path connection algorithm is probably the most widely used method for finding wire paths on printed circuit boards. It is shown that the original claim of generality for the path cost function is incorrect, and a restriction, called the pathconsistency property, is introduced. The Lee algorithm holds for those path cost functions having this property. Codings for the cells of the grid are proposed which will allow the correct operation of the algorithm under the most general path cost function, using the minimum number of states possible, six states per cell. Then methods for reducing the number of calculations by increasing the number of states are presented.

187 citations



Journal ArticleDOI

20 citations


Journal ArticleDOI
TL;DR: It is shown that a directed and weighted linear graph may be produced, on which this problem is equivalent to the well-known problem of finding the shortest path between a unique source and a unique sink.
Abstract: The problem dealt with in this paper is the optimal ordering of mother boards of a vast digital system, for instance, so that the total length of the interconnecting wires will be minimal. It is shown that a directed and weighted linear graph may be produced, on which this problem is equivalent to the well-known problem of finding the shortest path between a unique source and a unique sink. An algorithm for tracing the shortest path is presented. This algorithm takes advantage of the special features of the auxiliary graph and reduces significantly the computational effort required. Some upper bounds on the number of operations needed are included.

17 citations


Journal ArticleDOI
TL;DR: In this article, a sufficient degree-condition for the existence of k-rails in graphs is given, and the number of edges required in 3-connected graphs to guarantee the existence (5) of 5 k-rail is determined.

11 citations



Journal ArticleDOI
01 Jan 1974-Networks
TL;DR: Different types of directed and undirected alternating paths are defined, and it is shown how the problem of finding the shortest directed alternating path can be transformed into a problem ofFinding the shortest path in a directed graph.
Abstract: The concept of an alternating path has been very useful in analyzing matching problems and has formed the basis for a number of matching algorithms. However, no techniques have been devised to find the shortest alternating path in a weighted graph. This paper defines different types of directed and undirected alternating paths, and shows how the problem of finding the shortest directed alternating path can be transformed into a problem of finding the shortest path in a directed graph. Utilizing this transformation, an efficient algorithm is developed for finding the shortest undirected alternating path. Computational experience is given. Extensions of the techniques in this paper to other types of alternating paths are discussed.

8 citations