Showing papers on "Longest path problem published in 1985"

How to Find Long Paths Efficiently

Abstract: We study the complexity of finding long paths in directed or undirected graphs, Given a graph G =(V, E) and a number k our algorithm decides within time O(K! · |V| · |E|) for all u,v ɛ V Whether there exists some path of length k firm u to v. The complexity of this algorithm has to be compared with 0(|V| k−1 · |E|) Which is the worst case behaviour of the algorithms described up to now in the literature, We get similar results for the problems of finding a longest path, a cycle of length k or a longest cycle, respectively. Our approach is based on the idea of representing certain families of sets by subfamilies of small cardiality. We also discuss the border lines of this idea.

On the facial structure of scheduling polyhedra

Abstract: A well-known job shop scheduling problem can be formulated as follows. Given a graph G with node set N and with directed and undirected ares, find an orientation of the undirected ares that minimizes the length of a longest path in G. We treat the problem as a disjunctive, program, without recourse to integer, variables, and give a partial characterization of the scheduling polyhedron P(N), i.e., the convex hull of feasible schedules. In particular, we derive all the facet inducing inequalities for the scheduling polyhedron P(K) defined on some clique with node set K, and give a sufficient condition, for such inequalities to also induce facets of P(N). One of our results is that any inequality that induces a facet of P(H) for some H⊂K, also induces a facet of P(K). Another one is a characterization of adjacent facets in terms of the index sets of the nonzero coefficients of their defining inequalities. We also address the constraint identification problem, and give a procedure for finding an inequality that cuts off a given solution to a subset of the constraints.

Longest paths and cycles in K1,3-free graphs

Abstract: In this article we show that the standard results concerning longest paths and cycles in graphs can be improved for K1,3-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K1,3-free graphs.

On Shortest Paths in Graphs with Random Weights

Abstract: We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n/n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.

Critical path planning under uncertainty

Abstract: This paper concerns a CPM network in which individual job times are random variables. Specifically the time for each job consists of a component which is a linear function of the investment (up to some maximum) in that job and a random variable that is independent of the investment. It is desired to find the minimum investment required as a function of expected project completion time. The problem is solved by a cutting plane technique in which the investment allocations yield feasibility cuts. Because of the special structure of this problem, these cuts can be generated by solving a sequence of longest path problems in an acyclic network.

Planning the shortest path for a disc in O(n2log n) time

Abstract: Given a robot R, a set S of obstacles, and points p and q, the Shortest Path Problem is to find the shortest path for R to move from p to q without crashing into any of the obstacles. We show that if the problem is restricted to a disc-shaped robot in the plane with nonintersecting polygons as obstacles then the shortest path can be found in time O(n2log n) where n is the number of edges that make up the polygonal obstacles. This matches the best time currently known for the simpler problem of finding the shortest path in the plane for a point robot.

Rectilinear shortest paths with rectangular barriers

Abstract: We address ourselves to an instance of the Shortest Path problem with obstacles where a shortest path in the Manhattan (or L1) distance is sought between two points (source and destination) and the obstacles are n disjoint rectangles with sides parallel to the coordinate axes. A plane sweep technique is applied rather than the graph theoretic approach frequently used in the literature. We show that there has to be a path of minimum length between the two given points which is monotone in at least one of x or y directions. Then we present an algorithm of time complexity O(n log n) for constructing that path and show that our algorithm is optimal.Lastly, we address the query form of this problem in which given a source point and n obstacles, after O(n log n) time for preprocessing, a shortest path from the source point to a query point avoiding all the obstacles can be reported in O(t + log n) time, where t is the number of turns on the path.

The traveling salesman problem in graphs with 3-edge cutsets

Abstract: This paper analyzes decomposition properties of a graph that, when they occur, permit a polynomial solution of the traveling salesman problem and a description of the traveling salesman polytope by a system of linear equalities and inequalities. The central notion is that of a 3-edge cutset, namely, a set of 3 edges that, when removed, disconnects the graph. Conversely, our approach can be used to construct classes of graphs for which there exists a polynomial algorithm for the traveling salesman problem. The approach is illustrated on two examples, Halin graphs and prismatic graphs.

Estimating critical path and arc probabilities in stochastic activity networks

Abstract: This article describes a new procedure for estimating parameters of a stochastic activity network of N arcs. The parameters include the probability that path m is the longest path, the probability that path m is the shortest path, the probability that arc i is on the longest path, and the probability that arc i is on the shortest path. The proposed procedure uses quasirandom points together with information on a cutset ℋ of the network to produce an upper bound of O[(log K)N−|ℋ|+1/K] on the absolute error of approximation, where K denotes the number of replications. This is a deterministic bound and is more favorable than the convergence rate of 1/K1/2 that one obtains from the standard error for K independent replications using random sampling. It is also shown how series reduction can improve the convergence rate by reducing the exponent on log K. The technique is illustrated using a Monte Carlo sampling experiment for a network of 16 relevant arcs with a cutset of ℋ = 7 arcs. The illustration shows the superior performance of using quasirandom points with a cutset (plan A) and the even better performance of using quasirandom points with the cutset together with series reduction (plan B) with regard to mean square error. However, it also shows that computation time considerations favor plan A when K is small and plan B when K is large.

An all pairs shortest path algorithm with expected running time O(n 2logn)

Abstract: An algorithm is described that solves the all pairs shortest path problem for a nonnegatively weighted graph. The algorithm has an average requirement on quite general classes of random graphs of O(n2logn) time, where n is the number of vertices in the graph.

An algorithm for finding Hamilton cycles in random graphs

Abstract: This paper describes a polynomial time algorithm HAM that searches for Hamilton cycles in undirected graphs. On a random graph its asymptotic probability of success is that of the existence of such a cycle. If all graphs with n vertices are considered equally likely, then using dynamic programming on failure leads to an algorithm with polynomial expected time. Finally, it is used in an algorithm for solving the symmetric bottleneck travelling salesman problem with probability tending to 1, as n tends to ∞.

Technical Note-An Improved Implementation of Conditional Monte Carlo Estimation of Path Lengths in Stochastic Networks

Abstract: This note suggests an improvement to the Monte Carlo simulation technique of Sigal, Pritsker and Solberg for estimating the distribution of the shortest/longest path length in a stochastic network. This improvement also applies in network reliability estimation and PERT analysis.

Binary-decision graphs for implementation of Boolean functions

Abstract: Binary decision graphs provide the most efficient means for computing Boolean functions by program, and also relate to hardware implementation In the present paper, formal properties of binary decision graphs are considered, and rules are provided for the anticipation of path lengths and vertex requirements, which indicate computation time and program length, respectively Rules for the construction of efficient (but not truly optimal) binary decision graphs are provided, these rules are then applied in design examples

Event simulating system

Abstract: PURPOSE: To improve the calculation efficiency by representing a symmetrical system by a graph, deriving a long path in the graph, executing simulation, deriving only an influenced part as for a partial change of a condition, and updating the longest path. CONSTITUTION: For instance, in case of executing a train operating simulation, first of all, data such as a train diagram, an operating condition, an equipment condition, etc. are inputted, and a graph for representing a train operation is prepared. With respect to this graph, a simulation by the longest path system is executed, whether it is in a dead lock state or not is checked, and in case of a normal end, a forecasting result of the train operation by the simulation is outputted. On the other hand, in case when dead lock information is outputted, a changed data is inputted, and with respect to the changed data, the graph is corrected. By the simulation of the second time, whether it is by the longest path system or the parametric system is selected in accordance with the influence of a correction of the graph. In such a way, the calculation efficiency against the change of the simulation process and the partial condition of the system. COPYRIGHT: (C)1986,JPO&Japio

The least weight subsequence problem

Abstract: The least weight subsequence (LWS) problem is introduced, and is shown to be equivalent to the classic minimum path problem for directed graphs. A special case of the LWS problem is shown to be solvable in O(n log n) time generally and, for certain weight functions, in linear time. A number of applications are given, including an optimum paragraph formation problem and the problem of finding a minimum height B-tree, whose solutions realize improvement in asymptotic time complexity.

Algorithms solving path systems

Abstract: Al gorithms recognizing solvable path systems are presented and their time complexity is investigated. Among them there is an algorithm with a constant expected behaviour.