Distributed Approximation of Minimum Routing Cost Trees

We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph $G$ over $n$ nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a different (but short) message to each of its neighbors in each synchronous round. We provide a randomized $(2+\epsilon)$-approximation with runtime $O(D+\frac{\log n}{\epsilon})$ for unweighted graphs. Here, $D$ is the diameter of $G$. This improves over both, the (expected) approximation factor $O(\log n)$ and the runtime $O(D\log^2 n)$ of the best previously known algorithm. Due to stating our results in a very general way, we also derive an (optimal) runtime of $O(D)$ when considering $O(\log n)$-approximations as done by the best previously known algorithm. In addition we derive a deterministic $2$-approximation

Worst-Case Analysis of Network Design Problem Heuristics

The Optimal Network problem (as defined by Scott [16]) consists of selecting a subset of arcs that minimizes the sum of the shortest paths between all nodes subject to a budget constraint. This paper considers the worst-case behavior of heuristics for this prob'em. Let n be the number of nodes in the network and e be a constant between 0 and 1. For a general class of Optimal Network Problems, we show that the question of finding a solution which is always less than n times the optimal solution is NP-complete. This indicates that all polynomial-time heuristics for the problem most probably have poor worst-case performance. An upper bound for worst-case heuristic performance of 2n times the optimal solution is also derived. For a restricted version of the Optimal Network problem we describe a procedure whose maximum percentage of error is bounded by a constant.This research was supported, in part, by the U. S. Department of Transportation under Contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)

A Survey of Network Design Problems

Network design problems arise in many different application areas such as air freight, highway traffic, and communication systems. The intention of this survey is to present a coherent unified view of a number of papers in the network design literature. We discuss suggested solution procedures, computational experience, relations between various network models, and potential application areas. Promising topics of research for improving, solving, and extending the models reviewed in this survey are also indicated.Supported in part by the U.S. Department of Transportation under contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)

A biologically inspired network design model

A network design problem is to select a subset of links in a transport network that satisfy passengers or cargo transportation demands while minimizing the overall costs of the transportation. We propose a mathematical model of the foraging behaviour of slime mould P. polycephalum to solve the network design problem and construct optimal transport networks. In our algorithm, a traffic flow between any two cities is estimated using a gravity model. The flow is imitated by the model of the slime mould. The algorithm model converges to a steady state, which represents a solution of the problem. We validate our approach on examples of major transport networks in Mexico and China. By comparing networks developed in our approach with the man-made highways, networks developed by the slime mould, and a cellular automata model inspired by slime mould, we demonstrate the flexibility and efficiency of our approach

A biologically inspired network design model

A network design problem is to select a subset of links in a transport network that satisfy passengers or cargo transportation demands while minimizing the overall costs of the transportation. We propose a mathematical model of the foraging behaviour of slime mould P. polycephalum to solve the network design problem and construct optimal transport networks. In our algorithm, a traffic flow between any two cities is estimated using a gravity model. The flow is imitated by the model of the slime mould. The algorithm model converges to a steady state, which represents a solution of the problem. We validate our approach on examples of major transport networks in Mexico and China. By comparing networks developed in our approach with the man-made highways, networks developed by the slime mould, and a cellular automata model inspired by slime mould, we demonstrate the flexibility and efficiency of our approach

Approximation algorithms for the shortest total path length spanning tree problem

AbstractGiven an undirected graph with a nonnegative weight on each edge, the shortest total path length spanning tree problem is to find a spanning tree of the graph such that the total path length summed over all pairs of the vertices is minimized. In this paper, we present several approximation algorithms for this problem. Our algorithms achieve approximation ratios of 2, 15/8, and 3/2 in time O(n2+f(G)),O(n3), and O(n4) respectively, in which f(G) is the time complexity for computing all-pairs shortest paths of the input graph G and n is the number of vertices of G. Furthermore, we show that the approximation ratio of (4/3+ε) can be achieved in polynomial time for any constant ε>0

Computing Near-Optimal Solutions to the Steiner Problem in a Graph Using a Genetic Algorithm

A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bitstring encoding. A bitstring specifies selected Steiner vertices and the corresponding Steiner tree is computed using the Distance Network Heuristic. This scheme ensures that every bitstring correspond to a valid Steiner tree and thus eliminates the need for penalty terms in the cost function. The GA is tested on all SPG instances from the OR-Library of which the largest graphs have 2,500 vertices and 62,500 edges. When executed 10 times on each of 58 graph examples, the GA finds the global optimum at least once for 55 graphs and every time for 43 graphs. In total the GA finds the global optimum in 77 % of all program executions and is within 1 % from the global optimum in more than 92 % of all executions. The performance is compared to that of two branch-and-cut algorithms and one of the very best deterministic heuristics, an iterated version of the Shortest Path Heuristic (SPH-I). For all test examples but one, even the worst result ever found by the GA is equal to or better than the result of SPH-I and in many cases the average error ratio of the GA is an order of magnitude better than that of SPH-I. The runtime of the GA is moderate for all test examples. This is in contrast to SPH-I as well as the branch-and-cut algorithms, for which the runtime in some cases are extremely high

Developing Efficient Metaheuristics for Communication Network Problems by using Problem-specific Knowledge

Metaheuristics, such as evolutionary algorithms or simulated annealing, are widely applicable heuristic optimization strategies that have shown encouraging results for a large number of difficult optimization problems. To show high performance, metaheuristics need to be adapted to the properties of the problem at hand. This paper illustrates how efficient metaheuristics can be developed for communication network problems by utilizing problem-specific knowledge for the design of a high-quality problem representation. The minimum communication spanning tree (MCST) problem finds a communication spanning tree that connects all nodes and satisfies their communication requirements for a minimum total cost. An investigation into the properties of the problem reveals that optimum solutions are similar to the minimum spanning tree (MST). Consequently, a problem-specific representation, the link biased (LB) encoding, is developed, which represents trees as a list of floats. The LB encoding makes use of the knowledge that optimum solutions are similar to the MST, and encodes trees that are similar to the MST with a higher probability. Experimental results for different types of metaheuristics show that metaheuristics using the LB-encoding efficiently solve existing MCST problem instances from the literature, as well as randomly generated MCST problems of different sizes and types