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Minimum weight

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


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TL;DR: It is shown that the kMST problem is NP-hard even for points in the Euclidean plane, and a simple technique is used to provide a polynomiM-time solution for finding k-trees of minimum diameter.
Abstract: We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

187 citations

Journal ArticleDOI
TL;DR: In this article, a simple heuristic for determining the p-centre of a finite set of weighted points in an arbitrary metric space is described. But the computational effort is O(np) for an n-point set and the ratio of the objective function value of the heuristic solution to that of the optimum is bounded.

180 citations

Journal ArticleDOI
TL;DR: In this article, multifunctional sandwich panels with corrugated and prismatic diamond cores have been analyzed and their behavior compared with panels designed using truss and honeycomb cores.

167 citations

Journal ArticleDOI
01 May 1991-Networks
TL;DR: It is demonstrated that, in general, there exist cases in which no finite path is optimal leading us to define an infinite path in such a way that the minimum weight problem always has a solution.
Abstract: We investigate the minimum weight path problem in networks whose link weights and link delays are both functions of time. We demonstrate that, in general, there exist cases in which no finite path is optimal leading us to define an infinite path (naturally, containing loops) in such a way that the minimum weight problem always has a solution. We also characterize the structure of an infinite optimal path. In many practical cases, finite optimal paths do exist. We formulate a criterion that guarantees the existence of a finite optimal path and develop an algorithm to find such a path. Some special cases, e.g., optimal loopless paths, are also discussed.

165 citations

Journal ArticleDOI
TL;DR: This work considers the problem of constructing a minimum-weight, two-connected network spanning all the points in a set V and assumes a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality, and gets a structural characterization of optimal solutions.
Abstract: We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thatany two-connected graph satisfying these conditions is theunique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. We use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.

160 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202321
202239
202153
202051
201966
201858