The Polymatroid Steiner Problems

The Steiner tree problem asks for a shortest subgraph connecting a given set of terminals in a graph. It is known to be APX-complete, which means that no polynomial time approximation scheme can exist for this problem, unless P=NP. Currently, the best approximation algorithm for the Steiner tree problem has a performance ratio of 1.55, whereas the corresponding lower bound is smaller than 1.01. In this paper, we provide for several Steiner tree approximation algorithms lower bounds on their performance ratio that are much larger. For two algorithms that solve the Steiner tree problem on quasi-bipartite instances, we even prove lower bounds that match the upper bounds. Quasi-bipartite instances are of special interest, as currently all known lower bound reductions for the Steiner tree problem in graphs produce such instances.

We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph G and a root node r, we seek a single spanning tree T of minimum stretch, where the stretch of T is defined to be the maximum ratio, over all terminal sets X, of the cost of the minimal sub-tree T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</sub> of T that connects X to r to the cost of an optimal Steiner tree connecting X to r in G. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. graphs with 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(√log n)</sup> -stretch. We also give a polynomial time We provide a polynomial time UST construction for general polylog(n)-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first subpolynomial-stretch (o(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε</sup> ) for any ε >; 0) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs.

&lt;p&gt;Let G = (V,E) be an undirected simple graph and w : E -&gt; R+ be&lt;br /&gt;a non-negative weighting of the edges of G. Assume V is partitioned&lt;br /&gt;as R union X. A Steiner tree is any tree T of G such that every node&lt;br /&gt;in R is incident with at least one edge of T. The metric Steiner tree&lt;br /&gt;problem asks for a Steiner tree of minimum weight, given that w is a&lt;br /&gt;metric. When X is a stable set of G, then (G,R,X) is called quasi-bipartite.&lt;br /&gt; In [1], Rajagopalan and Vazirani introduced the notion of&lt;br /&gt;quasi-bipartiteness and gave a ( 3/2 + epsilon) approximation algorithm&lt;br /&gt; for the metric Steiner tree problem, when (G,R,X) is quasi-bipartite. In this&lt;br /&gt;paper, we simplify and strengthen the result of Rajagopalan and Vazirani.&lt;br /&gt;We also show how classical bit scaling techniques can be adapted&lt;br /&gt;to the design of approximation algorithms.&lt;/p&gt;&lt;p&gt;Key words: Steiner tree, local search, approximation algorithm, bit scaling.&lt;/p&gt;&lt;p&gt; &lt;/p&gt;

Boolean satisfiability (SAT) is a well-studied N P-complete problem for formulating and solving other combinatorial problems like planning and scheduling. The Steiner tree problem (STP) asks to find a minimal cost tree in a graph that spans a set of nodes. STP has been shown to be N P-hard. In this paper, we propose to solve the STP by formulating it as a variation of SAT, and to solve it using a heuristic search method guided by the backbone of the problem. The algorithm is tested on a well known set of benchmark instances. Experimental results demonstrate the applicability of the proposed approach, and show that substantial quality improvement can be obtained compared to other heuristic methods.

Given a connected graph G = (V ,E ) with nonnegative costs on edges, $c:E\rightarrow {\mathcal R}^+$, and a subset of terminal nodes R *** V , the Steiner tree problem asks for the minimum cost subgraph of G spanning R . The Steiner Tree Problem with distances 1 and 2 (i.e., when the cost of any edge is either 1 or 2) has been investigated for long time since it is MAX SNP-hard and admits better approximations than the general problem. We give a 1.25 approximation algorithm for the Steiner Tree Problem with distances 1 and 2, improving on the previously best known ratio of 1.279.

The Wireless sensor network has a great applied prospect, so the military, academia and industry pay more and more extensive attention. The first, this article introduces the concept of sensor networks and development background; the second, compared with traditional networks, it summarizes the characteristics of sensor networks. In this basis, the application of the sensor network and the direction of the current study are a given brief introduction and a further discussion.

Data aggregation reduces energy consumption by reducing the number of message transmissions in sensor networks. Effective aggregation requires that event messages be routed along common paths. While existing routing protocols provide many ways to construct the aggregation tree, this opportunistic style of aggregation is usually not optimal. The Minimal Steiner Tree (MST) maximises the possible degree of aggregation, but finding such a tree requires global knowledge of the network, which is not practical in sensor networks. In this paper, we propose the Adaptive Aggregation Tree (AAT) to dynamically transform the structure of the routing tree to improve the efficiency of data aggregation. It adapts to changes in the set of source nodes automatically, and approaches the cost savings of MST without explicit maintenance of an infrastructure. The evaluation results show that AAT reduces the communication energy consumption by 23%, compared to shortest-path tree, and by 31%, compared to GPSR.

This paper presents the first investigation on applying a particle swarm optimization (PSO) algorithm to both the Steiner tree problem and the delay constrained multicast routing problem. Steiner tree problems, being the underlining models of many applications, have received significant research attention within the meta-heuristics community. The literature on the application of meta-heuristics to multicast routing problems is less extensive but includes several promising approaches. Many interesting research issues still remain to be investigated, for example, the inclusion of different constraints, such as delay bounds, when finding multicast trees with minimum cost. In this paper, we develop a novel PSO algorithm based on the jumping PSO (JPSO) algorithm recently developed by Moreno-Perez et al. (Proc. of the 7th Metaheuristics International Conference, 2007), and also propose two novel local search heuristics within our JPSO framework. A path replacement operator has been used in particle moves to improve the positions of the particle with regard to the structure of the tree. We test the performance of our JPSO algorithm, and the effect of the integrated local search heuristics by an extensive set of experiments on multicast routing benchmark problems and Steiner tree problems from the OR library. The experimental results show the superior performance of the proposed JPSO algorithm over a number of other state-of-the-art approaches.

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

Efficient data collection in wireless sensor networks (SNs) plays a key role in power conservation. It has spurred a number of research projects focusing on effective algorithms that reduce power consumption with effective in-network aggregation techniques. Up to now, most approaches are based on the assumption that data collection involves all nodes of a network. There is a large number of queries that in fact select only a subset of the nodes in a SN. Thus, we concentrate on selective queries, i.e., queries that request data from a subset of a SN. The task of optimal data collection in such queries is an instance of the NP-hard minimal Steiner tree problem. We argue that selective queries are an important class of queries that can benefit from algorithms that are tailored for partial node participation of a SN. We present an algorithm, called Pocket Driven Trajectories (PDT), that optimizes the data collection paths by approximating the global minimal Steiner tree using solely local spatial knowledge. We identify a number of spatial factors that play an important role for efficient data collection, such as the distribution of participating nodes over the network, the location and dispersion of the data clusters, the location of the sink issuing a query, as well as the location and size of communication holes. In a series of experiments, we compare performance of well-known algorithms for aggregate query processing against the PDT algorithm in partial node participation scenarios. To measure the efficiency of all algorithms, we also compute a near-optimal solution, the globally approximated minimal Steiner tree. We outline future research directions for selective queries with varying node participation levels, in particular scenarios in which node participation is the result of changing physical phenomena as well as reconfigurations of the SN itself.KeywordsSensor NetworkSensor NodeWireless Sensor NetworkSteiner TreeSensor ReadingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Connecting a given set of points at minimum cost may be rated as one of the most important problems in telecommunications network design. Related questions may be formulated in metric spaces as well as in graphs. Given a weighted graph, the Steiner tree problem in graphs asks to determine a minimum cost subgraph spanning a set of specified vertices. This problem may be viewed as the combinatorial optimization problem in telecommunications. In this chapter, we survey Steiner problems from a telecommunications perspective with a special emphasis on the problem in graphs.

The rectilinear Steiner tree problem asks for a shortest network connecting n terminals in the rectilinear plane (with L1‐metric). The problem can be considered as a special case of the Steiner tree problem in a grid network obtained by drawing horizontal and vertical lines through terminals. The reductions developed for the network variant of the Steiner tree problem can therefore also be used in the rectilinear case. However, computational experience indicates that they are not very powerful in the rectilinear grids. We introduce a limited number of rectilinear reductions based on the geometrical properties of terminals. Experimental results show that these reductions are extremely powerful while being quite simple to verify and easy to implement. For randomly generated problem instances with one terminal per row and column, The number of nonterminals remaining seems to be around 20‐‐25% of their original number. Furthermore, The performance of geometrical reductions improves as the density of terminals in fixed‐size grids grows.

The energy problem is one of the most important challenges in the application of sensor networks. With the development of wireless charging technology and intelligent mobile charger (MC), the energy problem can be solved by the wireless charging strategy. In the practical application of wireless rechargeable sensor networks (WRSNs), the energy consumption rate of nodes is dynamically changed due to many uncertainties, such as the death and different transmission tasks of sensor nodes. However, existing works focus on on-demand schemes, which not fully consider real-time global charging scheduling. In this article, a novel dynamic charging scheme (DCS) in WRSN based on the actor-critic reinforcement learning (ACRL) algorithm is proposed. In the ACRL, we introduce gated recurrent units (GRUs) to capture the relationships of charging actions in time sequence. Using the actor network with one GRU layer, we can pick up an optimal or near-optimal sensor node from candidates as the next charging target more quickly and speed up the training of the model. Meanwhile, we take the tour length and the number of dead nodes as the reward signal. Actor and critic networks are updated by the error criterion function of R and V. Compared with current on-demand charging scheduling algorithms, extensive simulations show that the proposed ACRL algorithm surpasses heuristic algorithms, such as the Greedy, DP, nearest job next with preemption, and TSCA in the average lifetime and tour length, especially against the size and complexity increasing of WRSNs.

The General Steiner Tree-Star problem

Recently, the use of wireless sensor networks has spread to applications areas that are not viable or costefficient to be run on other types of networks. Due to some critical tasks done in these types of networks, the majority of sensor networks applications should be dependable and should be run continuously and reliably without interruption. Hence, the two more significant dependability factors that should be nowadays taken into account in developing wireless sensor networks applications are ‘availability’ and ‘reliability’ of application services. The specific characteristics and constraints of wireless sensor networks require a different interpretation of these two factors when developing applications for such networks. In this paper, we propose a middleware layer mechanism for satisfying these two factors as more important dependability issues in sensor networks applications. We propose an event-based middleware service that is specifically designed for wireless sensor networks in which a group of sensor nodes forms a cluster and a replicated service is run on each cluster head. The communication model among cluster members and cluster head is based on the publish/subscribe scheme. We show how the replicated services and communication model in cluster nodes satisfy dependability issues and increase the availability and reliability of applications running under the proposed middleware.