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

Let $G=(V,E)$ be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex $v\in V$ has a demand $d(v)\in Z_{+}$, and a cost $c(v)\in R_{+}$, where $Z_{+}$ and $R_{+}$ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing ${\sum }_{v\in S}c(v)$ such that there are at least $d(v)$ pairwise vertex-disjoint paths from S to v for each vertex $v\in V?S$. It is known that the problem is not approximable within a ratio of $O(\ln {\sum }_{v\in V}d(v))$, unless NP has an $O(N^{\log \log N})$-time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and $d^1=4$ holds, then the problem is NP-hard, where $d^1=\max \{d(v)|v\in V\}$.In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a $\max \{d^1,2d^1?6\}$-approximate solution to the problem in $O(\min \{d^1,\sqrt{|V|}\}{d}^1{|V|}^2)$ time, while we also show that there exists an instance for which it provides no better than a $(d^1?1)$-approximate solution. Especially, in the case of $d^1?4$, we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to $d^1?4$.     

Fault-tolerant concave facility location problem with uniform requirements

In this paper, we consider the fault-tolerant concave facility location problem (FTCFL) with uni- form requirements. By investigating the structure of the FTCFL, we obtain a modified dual-fitting bifactor approximation algorithm. Combining the scaling and greedy argumentation technique, the approximation fac- tor is proved to be 1.52.     

Source location with rigidity and tree packing requirements

We consider the following two problems: (i) given a graph, find a minimum size vertex-set such that the pinning of it makes the graph rigid in the plane. (ii) Given a graph, contract a minimum size vertex-set such that the resulting graph has two edge-disjoint spanning trees. We prove that these problems are polynomially solvable.     

An algorithm for source location in directed graphs

A polynomial time solution algorithm is described to find a smallest subset R of nodes of a directed graph D=(V,A) such that, for every node v∈V-R, there are k edge-disjoint paths from R to v and there are l edge-disjoint paths from v to R.     

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G=(V,E), a family W of sets W⊆V of vertices (areas), and a requirement function r:W→Z⁺ (where Z⁺ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W∈W. So far this problem was shown to be NP-hard in the uniform case of r(W)=1 for each W∈W, and polynomially solvable in the uniform case of r(W)=r?2 for each W∈W. In this paper, we show that the problem can be solved in time, even if r(W)?2 holds for each W∈W, where n=|V|, m=|{{u,v}|(u,v)∈E}|, p=|W|, and r^*=max{r(W)∣W∈W}.     

Augmenting forests to meet odd diameter requirements

Given a graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by the smallest number of new edges so that the diameter becomes at most D. It is known that no constant approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP. For a forest G and an odd D≥3, it was open whether the problem is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with a forest G and an odd D; our algorithm delivers an 8-approximate solution in O(|V|³) time. We also show that a 4-approximate solution to the problem with a forest G and an odd D can be obtained in linear time if the augmented graph is additionally required to be biconnected.     

Maximum-cover source location problems with objective edge-connectivity three

Given a graph G = (V, E), a set of vertices covers a vertex if the edge-connectivity between S and v is at least a given number k. Vertices in S are called sources. The maximum-cover source location problem, which is a variation of the source location problem, is to find a source set S with a given size at most p, maximizing the sum of the weight of vertices covered by S. In this paper, we show a polynomial-time algorithm for this problem in the case of k = 3 for a given undirected graph with a vertex weight function and an edge capacity function. Moreover, we show that this problem is NP-hard even if vertex weights and edge capacities are both uniform for general k.     

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

A generalized Weiszfeld method for the multi-facility location problem

An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities.     

A fast swap-based local search procedure for location problems

We present a new implementation of a widely used swap-based local search procedure for the p-median problem, proposed in 1968 by Teitz and Bart. Our method produces the same output as the best alternatives described in the literature and, even though its worst-case complexity is similar, it can be significantly faster in practice: speedups of up to three orders of magnitude were observed. We also show that our method can be easily adapted to handle the facility location problem and to implement related procedures, such as path-relinking and tabu search. R. F. Werneck: The results presented in this paper were obtained while this author was a summer intern at AT&T Labs Research.     

The central warehouse location problem revisited

This paper is concerned with the optimal location of a centralwarehouse, given a fixed number and the locations of the localwarehouses. We investigate whether the solution determined bythe traditional model that minimizes total transportation costdiffers from the one determined by a model that also takes intoaccount the inventory and service costs. We build simple modelsto address this question. Numerical results show that ignoringinventory costs in modelling location models may lead to inferiorlocation solutions.     

Two new heuristics for the location set covering problem

Summary In this paper we present two new greedy-type heuristics for solving the location set covering problem. We compare our new pair of algorithms with the pair GH1 and GH2 [Vasko and Wilson (1986)] and show that they perform better for a selected set of test problems.     

A maximumb-matching problem arising from median location models with applications to the roommates problem

We consider maximumb-matching problems where the nodes of the graph represent points in a metric space, and the weight of an edge is the distance between the respective pair of points. We show that if the space is either the rectilinear plane, or the metric space induced by a tree network, then theb-matching problem is the dual of the (single) median location problem with respect to the given set of points. This result does not hold for the Euclidean plane. However, we show that in this case theb-matching problem is the dual of a median location problem with respect to the given set of points, in some extended metric space. We then extend this latter result to any geodesic metric in the plane. The above results imply that the respective fractionalb-matching problems have integer optimal solutions. We use these duality results to prove the nonemptiness of the core of a cooperative game defined on the roommate problem corresponding to the above matching model. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

From stars to comets: Improved local search for universal facility location

We improve the approximation ratio for the Universal Facility Location Problem to 6.702 by a local search algorithm with an extended pivot operation.     

树状网络上的Web代理服务器最优放置问题

一般网络上Web代理服务器(Web proxy)最优放置问题是一个NP困难问题.此文讨论树状网络上的最优放置问题,改进了已有结果,得到了一个时间复杂度为O(nhk)的多项式时间算法,这里n为网络结点数,h为树的高度,而k为要放置的代理服务器个数.     

A note on optimality conditions for the Euclidean. Multifacility location problem

Thekey problem of the Euclidean multifacility location (EMFL) problem is to decide whether a givendead point is optimal. If it is not optimal, we wish to compute a descent direction. This paper extends the optimality conditions of Calamai and Conn and Overton to the case when the rows of the active constraints matrix are linearly dependent. We show that linear dependence occurs wheneverG, the graph of the coinciding facilities, has a cycle. In this case the key problem is formulated as a linear least squares problem with bounds on the Euclidean norms of certain subvectors.     

An approximation algorithm for the dynamic facility location problem with submodular penalties

In this paper, we study the dynamic facility location problem with submodular penalties （DFLPSP）. We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.     

Approximation algorithms for the stochastic priority facility location problem

In this article, we present a primal-dual 3-approximation algorithm for the stochastic priority facility location problem. Combined with greedy augmentation procedure, such performance factor is further improved to 1.8526.     

A Lagrangean heuristic for the maximal covering location problem

We develop a Lagrangean heuristic for the maximal covering location problem. Upper bounds are given by a vertex addition and substitution heuristic and lower bounds are produced through a subgradient optimization algorithm. The procedure was tested in networks of up to 150 vertices. A duality gap was generally present at the end of the heuristic for the larger problems. The test problems were run in an IBM 3090-600J ‘super-computer’; the maximum computing time was kept below three minutes of CPU.     

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.