Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms

A dual-based algorithm for multi-level network design

Includes bibliographical references.Supported in part by a grant from the AT&T Research Fund.Anantaram Balakrishnan, Thomas L. Magnanti, Prakash Mirchandani

Connectivity-splitting models for survivable network design

"January 2000." Title from cover.Includes bibliographical references (p. 24-25).by T.L. Magnanti, A. Balakrishnan, P. Mirchandani

A Dual-Based Algorithm for Multi-Level Network Design

Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher type. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem, and has applications in telecommunication, transportation, and electric power distribution network design. In a companion paper we introduced the problem, studied alternative model formulations, and analyzed the worst-case performance of heuristics based on Steiner network and spanning tree solutions. This paper develops and tests a dual-based algorithm for the Multi-level Network Design (MLND) problem. The method first performs problem preprocessing to fix certain design variables, and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value. We report extensive computational results on large, random networks (containing up to 500 nodes, and 5000 edges) with varying cost structures. The integer programming formulation of the largest of these problems has 20,000 integer variables and over 5 million constraints. Our tests indicate that the dualbased algorithm is very effective, producing solutions guaranteed to be within 0 to 0.9% of optimality

The node-weighted Steiner tree approach to identify elements of cancer-related signaling pathways

BACKGROUND Cancer constitutes a momentous health burden in our society. Critical information on cancer may be hidden in its signaling pathways. However, even though a large amount of money has been spent on cancer research, some critical information on cancer-related signaling pathways still remains elusive. Hence, new works towards a complete understanding of cancer-related signaling pathways will greatly benefit the prevention, diagnosis, and treatment of cancer. RESULTS We propose the node-weighted Steiner tree approach to identify important elements of cancer-related signaling pathways at the level of proteins. This new approach has advantages over previous approaches since it is fast in processing large protein-protein interaction networks. We apply this new approach to identify important elements of two well-known cancer-related signaling pathways: PI3K/Akt and MAPK. First, we generate a node-weighted protein-protein interaction network using protein and signaling pathway data. Second, we modify and use two preprocessing techniques and a state-of-the-art Steiner tree algorithm to identify a subnetwork in the generated network. Third, we propose two new metrics to select important elements from this subnetwork. On a commonly used personal computer, this new approach takes less than 2 s to identify the important elements of PI3K/Akt and MAPK signaling pathways in a large node-weighted protein-protein interaction network with 16,843 vertices and 1,736,922 edges. We further analyze and demonstrate the significance of these identified elements to cancer signal transduction by exploring previously reported experimental evidences. CONCLUSIONS Our node-weighted Steiner tree approach is shown to be both fast and effective to identify important elements of cancer-related signaling pathways. Furthermore, it may provide new perspectives into the identification of signaling pathways for other human diseases

Modeling and heuristic worst-case performance analysis of the two-level network design problem

"Revised: November 1992"--2nd prelim. p.Includes bibliographical references.Supported by a grant from the AT&T Research Fund. Supported by a Faculty Grant from the Katz Graduate School of Business, University of Pittsburgh.Anantaram Balakrishnan, Thomas L. Magnanti and Prakash Mirchandani

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