Minimum weight

About: Minimum weight is a research topic. Over the lifetime, 2002 publications have been published within this topic receiving 28244 citations.

Faster Parametric Shortest Path and Minimum Balance Algorithms

Abstract: The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents fast algorithms for both problems. The algorithms run in O(nm+n^2 log n) time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding a minimum-mean cycle (an important special case of both problems) is O(n log(n) + m).

Quasi-greedy triangulations approximating the minimum weight triangulation

Abstract: This article settles the following two longstanding open problems:?What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation??Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(nlogn) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.

Computing a minimum-weight k-link path in graphs with the concave Monge property

Abstract: LetGbe a weighted, complete, directed acyclic graph whose edge weights obey the concave Monge condition. We give an efficient algorithm for finding the minimum weightk-link path between a given pair of vertices for any givenk. The algorithm runs in time, fork=?(logn). Our algorithm can be applied to get efficient solutions for the following problems, improving on previous results: (1) computing length-limited Huffman codes, (2) computing optimal discrete quantization, (3) computing maximumk-cliques of an interval graph, (4) finding the largestk-gon contained in a given convex polygon, (5) finding the smallestk-gon that is the intersection ofkhalf-planes out ofnhalf-planes defining a convexn-gon.

A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships

Abstract: The sequential ordering problem with precedence relationships was introduced in Escudero [7]. It has a broad range of applications, mainly in production planning for manufacturing systems. The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the arcs, subject to precedence relationships among nodes. Nodes represent jobs (to be processed on a single machine), arcs represent sequencing of the jobs, and the weights are sums of processing and setup times. We introduce a formulation for the constrained minimum weight Hamiltonian path problem. We also define Lagrangian relaxation for obtaining strong lower bounds on the makespan, and valid cuts for further tightening of the lower bounds. Computational experience is given for real-life cases already reported in the literature.