We consider the following problem: given a set of points in the plane, each with a weight, and capacities of the four quadrants, assign each point to one of the quadrants such that the total weight of points assigned to a quadrant does not exceed its capacity, and the total distance is minimized.
This problem is most important in placement of VLSI circuits and is likely to have other applications. It is NP-hard, but the fractional relaxation always has an optimal solution which is “almost” integral. Hence for large instances, it suffices to solve the fractional relaxation. The main result of this paper is a linear-time algorithm for this relaxation. It is based on a structure theorem describing optimal solutions by so-called “American maps” and makes sophisticated use of binary search techniques and weighted median computations.
This algorithm is a main subroutine of a VLSI placement tool that is used for the design of many of the most complex chips. 相似文献
The aim of this paper is to propose an algorithm based on the philosophy of the Variable Neighborhood Search (VNS) to solve Multi Depot Vehicle Routing Problems with Time Windows. The paper has two main contributions. First, from a technical point of view, it presents the first application of a VNS for this problem and several design issues of VNS algorithms are discussed. Second, from a problem oriented point of view the computational results show that the approach is competitive with an existing Tabu Search algorithm with respect to both solution quality and computation times. 相似文献
Let F be a function field of characteristic p > 2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F. 相似文献
Zusammenfassung. Optimale Quantisierungen oder – damit ?quivalent – minimale Summen von Momenten spielen in mehreren Zweigen der Mathematik
und ihrer Anwendungen eine Rolle. Ausgehend von der Fejes Tóth'schen Ungleichung für Summen von Momenten in der euklidischen
Ebene und einem zugeh?rigen Stabilit?tssatz, werden gewisse Erweiterungen auf normierte R?ume und riemannsche Mannigfaltigkeiten
h?herer Dimension besprochen. Die Ergebnisse werden dann auf Probleme aus folgenden Bereichen angewendet: (i) Datenübertragung,
(ii) Wahrscheinlichkeitstheorie, (iii) numerische Integration, (iv) Approximation konvexer K?rper und (v) isoperimetrische
Probleme.
Eingegangen am 29. Mai 2002 / Angenommen am 8. Juli 2002 相似文献
Where there is abundance of mystery and confusion in every direction, the truth seldom remains hidden for long. It's a matter of having plenty of angles to go at it from. Only the utterly simple crimes - the simplex crimes, you may say - have the trick of remaining baffling. - Sir John (from Michael Innes,The Open House (A Sir John Appleby Mystery), Penguin Books, 1974).A dual simplex method for the assignment problem leaves open to choice the activity (i,j) of rowi and columnj that is to be dropped in pivoting so long asxij < 0. A choice (i,j) over columnsj having at least 3 basic activities that minimizesxij is shown to converge in at most (
2n-1
) pivots, and at most O(n3) time, and it is argued that on average the number of pivots is at mostn logn.
Dedicated with affection to George B. Dantzig on the occasion of his seventieth birthday. 相似文献
Christofides [1] proposes a heuristic for the traveling salesman problem that runs in polynomial time. He shows that when the graphG = (V, E) is complete and the distance matrix defines a function onV × V that is metric, then the length of the Hamiltonian cycle produced by the heuristic is always smaller than 3/2 times the length of an optimal Hamiltonian cycle. The purpose of this note is to refine Christofides' worst-case analysis by providing a tight bound for everyn 3, wheren is the number of vertices of the graph. We also show that these bounds are still tight when the metric is restricted to rectilinear distances, or to Euclidean distances for alln 6.This work was supported, in part, by NSF Grant ENG 75-00568 to Cornell University. This work was done when the authors were affiliated with the Center for Operations Research and Econometrics, University of Louvain, Belgium. 相似文献
A variable dimension algorithm is presented for the linear complementarity problems – Mz = q; s,z 0; sizi = 0 fori = 1,2, ,n. The algorithm solves a sequence of subproblems of different dimensions, the sequence being possibly nonmonotonic in the dimension of the subproblem solved. Every subproblem is the linear complementarity problem defined by a leading principal minor of the matrixM. Index-theoretic arguments characterize the points at which nonmonotonic behavior occurs. 相似文献