A linear time algorithm for the reverse 1‐median problem on a cycle

Abstract

This article deals with the reverse 1-median problem on graphs with positive vertex weights. The problem is proved to be strongly N P-hard even in the case of bipartite graphs and not approximable within a constant factor (unless P = N P). Furthermore, a linear time algorithm for the reverse 1-median problem on a cycle with linear cost functions (RMC) is developed. It is also shown that there exists an integral optimal solution of RMC if the input data are integral.

Upgrading p-Median Problem on a Path

“…Case II: There exists one index j ∈ J (2) k with x * j > 0. Let t be the smallest index in J (2) k so that x * t > 0 and s the smallest index in J (1) k so that x * s < u s .…”

“…Reverse median problems have been investigated by Berman et al [1] who discussed a reverse 1-median problem on a special graph, namely on a tree with linear cost functions. In [2], Burkard et al proved reverse 1-median problem with variable edge lengths is N P-hard. They consider reverse 1-median problem with a unit cost function on cycle graphs [2] and reverse 2-median problem [3].…”

“…In [2], Burkard et al proved reverse 1-median problem with variable edge lengths is N P-hard. They consider reverse 1-median problem with a unit cost function on cycle graphs [2] and reverse 2-median problem [3].…”

The inverse Fermat–Weber problem

“…For trees with n vertices they derived an algorithm with O(n log n) time complexity. On the other hand, Burkard, Gassner and Hatzl [2] proved that the reverse 1-median problem is N P-hard on general graphs but can be solved in linear time on a cycle. In [3] the same authors suggest an O(n log n) time algorithm for the reverse 2-median problem on trees and the reverse 1-median problem on unicycle graphs.…”

An Efficient Algorithm for Reverse 2-median Problem on A Cycle

“…This kind of problems is also called the network improvement problems or the reverse location problems. The reverse 1-median problem as well as the reverse 2median problem on general graphs are known to be strongly NP-hard ( [3], [4]). Therefore, special networks have been studied.…”