The Single Period Coverage Facility Location Problem: Lagrangean heuristic and column generation approaches

In this paper we introduce the Single Period Coverage Facility Location Problem. It is a multi-period discrete location problem in which each customer is serviced in exactly one period of the planning horizon. The locational decisions are made independently for each period, so that the facilities that are open need not be the same in different time periods. It is also assumed that at each period there is a minimum number of customers that can be assigned to the facilities that are open. The decisions to be made include not only the facilities to open at each time period and the time period in which each customer will be served, but also the allocation of customers to open facilities in their service period. We propose two alternative formulations that use different sets of decision variables. We prove that in the first formulation the coefficient matrix of the allocation subproblem that results when fixing the facilities to open at each time period is totally unimodular. On the other hand, we also show that the pricing problem of the second model can be solved by inspection. We prove that a Lagrangean relaxation of the first one yields the same lower bound as the LP relaxation of the second one. While the Lagrangean dual can be solved with a classical subgradient optimization algorithm, the LP relaxation requires the use of column generation, given the large number of variables of the second model. We compare the computational burden for obtaining this lower bound through both models

Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic

The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects''. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky. This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagriangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete Algorithms (1998

Robust capacitated trees and networks with uniform demands

We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in case of terminals with uniform demands. Formally, we are given a graph, capacity and cost functions on the edges, a root, a subset of nodes called terminals, and a bound k on the number of edge failures. We first study the problem where k = 1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Second, we consider the problem of computing a minimum-cost survivable network, i.e., a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set based formulation, a flow based one, and a bilevel one (with an attacker and a defender). We propose algorithms to solve each formulation and compare their efficiency

A new unifying heuristic algorithm for the undirected minimum cut problems using minimum range cut algorithms

AbstractGiven a connected undirected multigraph with n vertices and m edges, we first propose a new unifying heuristic approach to approximately solving the minimum cut and the s-t minimum cut problems by using efficient algorithms for the corresponding minimum range cut problems. Our method is based on the association of the range value of a cut and its cut value when each edge weight is chosen uniformly randomly from the fixed interval. Our computational experiments demonstrate that this approach produces very good approximate solutions. We shall also propose an O(log2 n) time parallel algorithm using O(n2) processors on an arbitrary CRCW PRAM model for the minimum range cut problems, by which we can efficiently obtain approximate minimum cuts in poly-log time using a polynomial number of processors

Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;