Book ChapterDOI
On the Uncapacitated Location Problem
TLDR
In this paper, a Lagrangian dual for obtaining an upper bound and heuristics for obtaining a lower bound on the value of an optimal solution are introduced, and the main results are analytical worst case analyses of these bounds.Abstract:
The problem of optimally locating bank accounts to maximize clearing times in discused. The importance of this problem depends in part on its mathematical relationship to the well-known uncapacitated plant location problem. A Lagrangian dual for obtaining an upper bound and heuristics for obtaining a lower bound on the value of an optimal solution are introduced. The main results are analytical worst case analyses of these bounds. In particular it is shown that the relative error of the dual bound and a “greedy” heuristic never exceeds [( K – 1)/ K ] K e for a problem in which at most K locations are to be chosen. An interchange heuristic is shown to have a worst case relative error of ( K – 1)/(2 K – 1)read more
Citations
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Journal ArticleDOI
The simple plant location problem: Survey and synthesis
Jakob Krarup,Peter Pruzan +1 more
TL;DR: In this paper, the authors consider a family of discrete, deterministic, single-criterion, NP-hard problems, including set packing, set covering, and set partitioning.
Posted Content
Learning with Submodular Functions: A Convex Optimization Perspective
TL;DR: Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular function and (2) the lovasz extension of sub-modular Functions provides a useful set of regularization functions for supervised and unsupervised learning as discussed by the authors.
Proceedings ArticleDOI
A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization
TL;DR: This work presents a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al.
Book
Learning with Submodular Functions: A Convex Optimization Perspective
TL;DR: In Learning with Submodular Functions: A Convex Optimization Perspective, the theory of submodular functions is presented in a self-contained way from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems.
Proceedings ArticleDOI
Submodular maximization with cardinality constraints
TL;DR: Improved approximations for two variants of the cardinality constraint for non-monotone functions are presented and a simple randomized greedy approach is presented where in each step a random element is chosen from a set of "reasonably good" elements.
References
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Journal ArticleDOI
Approximation algorithms for combinatorial problems
TL;DR: For the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as n^@e, where n is the problem size and @e>0 depends on the algorithm.
Journal ArticleDOI
The Traveling-Salesman Problem and Minimum Spanning Trees
Michael Held,Richard M. Karp +1 more
TL;DR: It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.
Book ChapterDOI
Lagrangian Relaxation for Integer Programming
TL;DR: It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8].
Journal ArticleDOI
The traveling-salesman problem and minimum spanning trees: Part II
Michael Held,Richard M. Karp +1 more
TL;DR: An efficient iterative method for approximating this bound closely from below is presented, and a branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it.
Journal ArticleDOI
On the Computational Complexity of Combinatorial Problems
TL;DR: A large class of classical combinatorial problems, including most of the difficult problems in the literature of network flows and computational graph theory, are shown to be equivalent, in the sense that either all or none of them can be solved in polynomial time.