Minimum Cost Source Location Problems with Flow Requirements

Abstract

In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ andκ), where the source location problems are to find a minimum-cost set S ⊆ V in a given graph G = (V , A) with a capacity function u : A → R + such that for each vertex v ∈ V , the connectivity from S to v (resp., from v to S) is at least a given demand d − (v) (resp., d + (v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54-68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of c ln D for some constant c, unless every problem in NP has an O(N log log N )-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1 + ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are (ln v∈V (d + (v) + d − (v)))-approximable.

Sink location to find optimal shelters in evacuation planning

“…Since the plural, non-simultaneous, independent sink location problems is already strongly NP-hard [14], this remains true for the plural, non-simultaneous, additive sink location problems.…”

“…However, these algorithms are only applicable to instances where the sources can be ordered in a specific way. N P-hardness for the general problem was initially shown by Arata et al [3], and Sakashita et al [14] showed that it is even N P-hard in the strong sense. The adaptation of the O(nM (n, m))-algorithm to the n|max|0 sink location problem is stated as Algorithm 2.…”

Greedy approximation for the source location problem with vertex-connectivity requirements in undirected graphs

“…Also, Tamura et al [18] showed that the case of uniform costs and general demands is solvable in polynomial time, while the fastest known algorithm for it achieves complexity O (mM(n, m)) due to Arata et al [2], where n = |V |, m = |{{u, v} | u, v ∈ V }|, and M(n, m) denotes the time for max-flow computation in the graph with n vertices and m edges. In general, Sakashita et al [16] showed that the problem is strongly NP-hard. It is also known that when a given graph is a tree, the problem is weakly NP-hard [2] and there exists a pseudo-polynomial time algorithm for it [11,16].…”

“…In general, Sakashita et al [16] showed that the problem is strongly NP-hard. It is also known that when a given graph is a tree, the problem is weakly NP-hard [2] and there exists a pseudo-polynomial time algorithm for it [11,16].…”

Minimum Degree Orderings

“…1(a), where the numbers beside edges indicate their weights and each of the nontrivial extreme subsets X 1 , X 2 , X 3 ∈ X (G, w) is depicted by a dotted closed curve; (b) The tree representation for X (G, w) by Watanabe and Nakamura [28] to solve the edge-connectivity augmentation problem. Extreme subsets of graphs are an important tool to design efficient algorithms for solving graph connectivity problems such as the source location problem [16,24], the minimum k-way cut problem [21], and the dynamic minimum cut problem [17] in addition to the connectivity augmentation problem. Alia and Maestrini [1], Borgatti et al [4], and Naor et al [22] showed that extreme sets of a graph can be constructed efficiently from a Gomory-Hu tree of the graph.…”