Collection depots facility location problems in trees

Abstract

We consider a generalization of the median and center facility location problem called the collection depots facility location (CDFL) problem. We are given a set of client locations and a set of collection depots and we are required to find the placement for a certain number of facilities, so that the cost of dispatching a vehicle from a facility, to a client, to a collection depot, and back, is optimized for all clients. The CDFL center problem minimizes the cost of the most expensive vehicle tour among all clients, and the CDFL median problem minimizes the sum of the tour costs for all clients. We provide the first polynomial time algorithms to solve the 1 and k median problems in trees with time complexities O(n log n) and O(kn 3 ), respectively, where n is the number of vertices in the tree. In contrast, a restricted version of the k -median problem, where clients are given lists of allowed collection depots, is NP-complete even for star graphs. We also give an optimal linear time algorithm to solve the discrete and continuous weighted 1-center problem, improving on the O(n log n) result of Tamir and Halman [Discrete Optimization 2(2005), 168-184].

Improved Minmax Regret 1-Center Algorithms for Cactus Networks with c Cycles

“…If the given tree is not balanced and binary, Lemma 4 is still valid, if we use spine tree decomposition [5,6,7]. Thus, Theorem 2 and Lemma 4 imply …”

“…We assume that T is a balanced binary tree, hence its height is O(log n). (If not, algorithmic properties and the tools developed here for the balanced trees can easily be extended, using spine tree decomposition [5,6,7].) As in Sec.…”

“…We also assume that T is balanced. If not, spine tree decomposition [5,6,7], can convert it into a structure that has properties of a balanced binary tree in linear time. Let T (v) denote the subtree of T rooted at vertex v. For each vertex v i ∈ V , we introduce the upper envelope for the cost functions of the vertices in T (v i ) under s 0 by…”

“…Otherwise, we can insert dummy vertices of weight 0 and dummy edges of length 0. A subgraph that hangs from vertex u ∈ C, excluding u and the edge to u, is called a graft, 5 and is denoted by Γ (u). See Fig.…”

On the Round-Trip 1-Center and 1-Median Problems

“…When the number of servers p is an arbitrary positive integer, the classical p-center and p-median problems on general graphs are NP-hard [11], so are the respective round-trip and one-way location problems. Efficient algorithms for the collection depots problems exist for the case p = 1 [1,8,13], for the case |A i | = 1 [8,13], or for special classes of graphs [1,2,4,8,13]. Efficient approximation algorithms can be found in [9,13].…”