An inverse approach to convex ordered median problems in trees

Abstract

The convex ordered median problem is a generalization of the median, the k-centrum or the center problem. The task of the associated inverse problem is to change edge lengths at minimum cost such that a given vertex becomes an optimal solution of the location problem, i.e., an ordered median. It is shown that the problem is NP-hard even if the underlying network is a tree and the ordered median problem is convex and either the vertex weights are all equal to 1 or the underlying problem is the k-centrum problem. For the special case of the inverse unit weight k-centrum problem a polynomial time algorithm is developed.

Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

“…Consider the value ∆(L, R) defined by (11). On every path P sz let s 1 ( = s), s 2 , ..., s i(z) be the sequence of vertices from s to z with the following three properties…”

“…This property allows the inverse 1-median problem to be solved on a path with negative weights in O(n) time. Gassner [11] considered an inverse version of the convex ordered median problem and showed that this problem is N P-hard on general graphs, even on trees. Further, it was shown that the problem remains N P-hard for unit weights or if the underlying problem is a k-centrum problem (but not, if both of these conditions hold).…”

The inverse convex ordered 1-median problem on trees under Chebyshev norm and Hamming distance

“…Otherwise, the problem can be solved in quadratic time. On the other hand, the first who studied the inverse ordered median location problem on trees A C C E P T E D M A N U S C R I P T was Gassner [11]. She proved that the inverse convex ordered median problem on trees with uniform weights is NP-hard, the inverse k-centrum problem on a general weighted tree is also NP-hard.…”

Inverse p-median problems with variable edge lengths

“…The author suggested an O(n log n) time algorithm for this problem on a tree. An inverse version of the convex ordered median problem was studied by Gassner [11]. The author showed that this problem is N P-hard even on trees.…”