On the terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ+2, where ρ is the best-known approximation ratio for the graph Steiner tree problem.     

On approximation algorithms for the terminal Steiner tree problem

The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1−o(1))lnn unless NP has slightly superpolynomial time algorithms. Moreover, we present a polynomial time approximation algorithm for the metric version of this problem with a performance ratio of 2ρ, where ρ denotes the best known approximation ratio for the Steiner tree problem. This improves the previously best known approximation ratio for the metric terminal Steiner tree problem of ρ+2.     

On the partial terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = ( V,E ) with length function l:E → R ⁺ and two vertex subsets R ⊂ V and R ′ ⊆ R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R ∖ R ′ belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths ∑ _{(u,v) ∈T} l ( u,v ) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem.

Reoptimization of Steiner trees: Changing the terminal set

Given an instance of the Steiner tree problem together with an optimal solution, we consider the scenario where this instance is modified locally by adding one of the vertices to the terminal set or removing one vertex from it. In this paper, we investigate the problem of whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. Our results are as follows: (i) We prove that these reoptimization variants of the Steiner tree problem are NP-hard, even if edge costs are restricted to values from {1,2}$\{1,2\}$. (ii) We design 1.5-approximation algorithms for both variants of local modifications. This is an improvement over the currently best known approximation algorithm for the classical Steiner tree problem which achieves an approximation ratio of 1+ln(3)/2≈1.55$1+ln\left(3\right)/2\approx 1.55$. (iii) We present a PTAS for the subproblem in which the edge costs are natural numbers {1,…,k}$\{1,\dots ,k\}$ for some constant k$k$.     

The General Steiner Tree-Star problem

The Steiner tree problem is defined as follows—given a graph G=(V,E) and a subset X⊂V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a “switch” cost in addition to the cost of the edges. Switches are placed at internal nodes of the tree (essentially, we may assume that all non-leaf nodes of the Steiner tree have a switch). The cost for placing a switch may vary from node to node. A restricted version of this problem, where the terminal set X cannot be connected to each other directly but only via the Steiner nodes V?X, is referred to as the Steiner Tree-Star problem. The General Steiner Tree-Star problem does not require the terminal set and Steiner node set to be disjoint. This generalized problem can be reduced to the node weighted Steiner tree problem, for which algorithms with performance guarantees of Θ(lnn) are known. However, such approach does not make use of the fact that the edge costs form a metric. In this paper we derive approximation algorithms with small constant factors for this problem. We show two different polynomial time algorithms with approximation factors of 5.16 and 5.     

Elementary approximation algorithms for prize collecting Steiner tree problems

This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G=(V,E), a collection T={T₁,…,T_k}, each a subset of V of size at least 2, a weight function , and a penalty function . The goal is to find a forest F that minimizes the cost of the edges of F plus the penalties paid for subsets T_i whose vertices are not all connected by F. Our main result is a -approximation for the prize collecting generalized Steiner forest problem, where n2 is the number of vertices in the graph. This obviously implies the same approximation for the special case called the prize collecting Steiner forest problem (all subsets T_i are of size 2). The approximation algorithm is obtained by applying the local ratio method, and is much simpler than the best known combinatorial algorithm for this problem.Our approach gives a -approximation for the prize collecting Steiner tree problem (all subsets T_i are of size 2 and there is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [M.X. Goemans, D.P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing 24 (2) (April 1995) 296–317]. Another special case of our main algorithm is Bar-Yehuda's local ratio -approximation for the generalized Steiner forest problem (all the penalties are infinity) [R. Bar-Yehuda, One for the price of two: a unified approach for approximating covering problems, Algorithmica 27 (2) (June 2000) 131–144]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda.     

A 3.4713-approximation algorithm for the capacitated multicast tree routing problem

Given an underlying communication network represented as an edge-weighted graph G=(V,E), a source node sV, a set of destination nodes DV, and a capacity k which is a positive integer, the capacitated multicast tree routing problem asks for a minimum cost routing scheme for source s to send data to all destination nodes, under the constraint that in each routing tree at most k destination nodes are allowed to receive the data copies. The cost of the routing scheme is the sum of the costs of all individual routing trees therein. Improving on our previous approximation algorithm for the problem, we present a new algorithm which achieves a worst case performance ratio of , where ρ denotes the best known approximation ratio for the Steiner minimum tree problem. Since ρ is about 1.55 at the writing of the paper, the ratio achieved by our new algorithm is less than 3.4713. In comparison, the previously best ratio was .     

Primal-dual approximation algorithms for the Prize-Collecting Steiner Tree Problem

The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2−1/(n−1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n³logn) time—it applies the primal-dual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n²logn), as it applies this scheme only once, and achieves the slightly better ratio of (2−2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.     

A note on the minimum label spanning tree

We give a tight analysis of the greedy algorithm introduced by Krumke and Wirth for the minimum label spanning tree problem. The algorithm is shown to be a (ln(n−1)+1)-approximation for any graph with n nodes (n>1), which improves the known performance guarantee 2lnn+1.     

Steiner trees in uniformly quasi-bipartite graphs

The area of approximation algorithms for the Steiner tree problem in graphs has seen continuous progress over the last years. Currently the best approximation algorithm has a performance ratio of 1.550. This is still far away from 1.0074, the largest known lower bound on the achievable performance ratio. As all instances resulting from known lower bound reductions are uniformly quasi-bipartite, it is interesting whether this special case can be approximated better than the general case. We present an approximation algorithm with performance ratio 73/60<1.217 for the uniformly quasi-bipartite case. This improves on the previously known ratio of 1.279 of Robins and Zelikovsky. We use a new method of analysis that combines ideas from the greedy algorithm for set cover with a matroid-style exchange argument to model the connectivity constraint. As a consequence, we are able to provide a tight instance.     

A factoring approach for the Steiner tree problem in undirected networks

In communication networks, many applications, such as video on demand and video conferencing, must establish a communications tree that spans a subset K in a vertex set. The source node can then send identical data to all nodes in set K along this tree. This kind of communication is known as multicast communication. A network optimization problem, called the Steiner tree problem (STP), is presented to find a least cost multicasting tree. In this paper, an O(|E|) algorithm is presented to find a minimum Steiner tree for series-parallel graphs where |E| is the number of edges. Based on this algorithm, we proposed an O(2^2c·|E|) algorithm to solve the Steiner tree problem for general graphs where c is the number of applied factoring procedures. The c value is strongly related to the topology of a given graph. This is quite different from other algorithms with exponential time complexities in |K|.     

基于CBT的多源Steiner树构造算法

构造最小代价树问题可形式化为图论中Steiner树问题。而Steiner树的求解已经被证明是一个NP-complete问题,不可能在多项式时间求得其精确解,所以出现许多启发式算法：在可接受时间内,得到一棵近似的最优多播树。这些算法一般先指定所有链接边的费用,通过一定方法或规则,找出包含源端和所有目的端的一棵近似最优的多播树。很显然,它们并没有考虑由于路径的共享重叠而引起最小生成树链接边费用的变化。现利用CBT算法思想对变化的费用进行建模并对典型启发式算法作了改进,以适应不断变化了的链路费用。     

Co-evolutionary algorithms to solve hierarchized Steiner tree problems in telecommunication networks

This study investigates a hierarchized Steiner tree problem in telecommunication networks. In such networks, users must be connected to capacitated hubs. Additionally, selected hubs must be connected to each other and to extra hubs, if necessary, by considering the latency of the resultant network. A connection between hubs can be considered to be a Steiner tree. This Steiner tree problem is modeled as a bi-level mathematical programming problem that considers two decision levels. In the upper-level, the allocation of users to hubs is performed to minimize the total network connection cost. The lower-level minimizes the user latency concerning the information that flows through the capacitated hubs. Further, two co-evolutionary schemes are developed to solve this bi-level model. The first scheme is an individual–population approach, whereas the second scheme is the traditional population–population approach. The first proposed algorithm exploits the structure of the problem by employing parallel computing in one of the populations. Numerical results depict the effectiveness of the proposed algorithms when the lower-level problem cannot be optimally solved efficiently. Furthermore, the advantages of the proposed schemes over an evolutionary one are exhibited. Finally, the hybridization of both co-evolutionary schemes is implemented to improve the semi-feasible solutions obtained by the second scheme, showing its effectiveness to solve the problem.     

物流网络中节点带权的Steiner最小树的参数算法

通过优化物流的运输网络,可以有效地降低物流成本。集中配送的物流网络优化问题可以转换成求解节点带权的Steiner最小树问题,这是一个NP-hard问题。运用参数理论,提出一种新的启发式解决算法P-NSMT。算法的思想是：首先尽可能只利用终端节点构造一棵连通的最小生成树,然后逐步向树中添加能减少生成树总权值的Steiner节点,最终生成一棵节点总数不超过参数k的Steiner最小树。实验表明,与同类型其他算法相比,P-NSMT算法具有更好的准确性和时间效率,特别适应于网络规模大、终端配送节点数目较少的物流网络。     

A hybrid Lagrangian genetic algorithm for the prize collecting Steiner tree problem

We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset of nodes and collecting a total prize not less that a given quota Q.$Q.$ We present a lower bound and a genetic algorithm for the PCSTP. The lower bound is based on a Lagrangian decomposition of a minimum spanning tree formulation of the problem. The volume algorithm is used to solve the Lagrangian dual. The genetic algorithm incorporates several enhancements. In particular, it fully exploits both primal and dual information produced by Lagrangian decomposition. The proposed lower and upper bounds are assessed through computational experiments on randomly generated instances with up to 500 nodes and 5000 edges. For these instances, the proposed lower and upper bounds exhibit consistently a tight gap: in 76% of the cases the gap is strictly less than 2%.     

新的基于MPH的时延约束Steiner树算法

为了在时延约束条件下进一步优化多播树代价并降低算法的复杂度,研究了时延受限的Steiner树问题。在DCMPH算法的基础上,通过改进节点的搜索路径,提出了一种新的基于MPH的时延约束Steiner树算法。该算法中每个目的节点通过最小代价路径加入当前多播树;若时延不满足要求,则通过合并最小时延树进而产生一个满足时延约束的最小代价多播树。仿真实验表明,新算法在性能、空间复杂度方面均优于DCMPH算法。     

New primal–dual algorithms for Steiner tree problems

We present new primal–dual algorithms for several network design problems. The problems considered are the generalized Steiner tree problem (GST), the directed Steiner tree problem (DST), and the set cover problem (SC) which is a subcase of DST. All our problems are NP-hard; so we are interested in their approximation algorithms. First, we give an algorithm for DST which is based on the traditional approach of designing primal–dual approximation algorithms. We show that the approximation factor of the algorithm is k, where k is the number of terminals, in the case when the problem is restricted to quasi-bipartite graphs. We also give pathologically bad examples for the algorithm performance. To overcome the problems exposed by the bad examples, we design a new framework for primal–dual algorithms which can be applied to all of our problems. The main feature of the new approach is that, unlike the traditional primal–dual algorithms, it keeps the dual solution in the interior of the dual feasible region. The new approach allows us to avoid including too many arcs in the solution, and thus achieves a smaller-cost solution. Our computational results show that the interior-point version of the primal–dual most of the time performs better than the original primal–dual method.     

Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-84-0140. Reproduction in whole or part is permitted for any purpose of the United States Government.     

On exact solutions to the Euclidean bottleneck Steiner tree problem

We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edge in the tree is minimized. This problem is known to be NP-hard even to approximate within ratio and there was no known exact algorithm even for k=1 prior to this work. In this paper, we focus on finding exact solutions to the problem for a small constant k. Based on geometric properties of optimal location of Steiner points, we present an optimal -time exact algorithm for k=1 and an O(n²)-time algorithm for k=2. Also, we present an optimal -time exact algorithm for any constant k for a special case where there is no edge between Steiner points.     

A note on the greedy algorithm for the unsplittable flow problem

In a recent paper Chekuri and Khanna improved the analysis of the greedy algorithm for the edge disjoint paths problem and proved the same bounds also for the related uniform capacity unsplittable flow problem. Here we show that their ideas can be used to get the same approximation ratio even for the more general Unsplittable Flow Problem with nonuniform edge capacities.