Algorithms For Network Design Problems Research Articles

A Spectral Approach to Network Design

We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional non-negative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem, and partially answers a question of Bansal about survivable network design with concentration property. We also show many other applications of the spectral rounding results, including weighted experimental design and additive spectral sparsification.

Development and Comparison of Hybrid Genetic Algorithms for Network Design Problem in Closed Loop Supply Chain

This paper presents four different hybrid genetic algorithms for network design problem in closed loop supply chain. They are compared using a complete factorial experiment with two factors, viz. problem size and algorithm. Based on the significance of the factor “algorithm”, the best algorithm is identified using Duncan’s multiple range test. Then it is compared with a mathematical model in terms of total cost. It is found that the best hybrid genetic algorithm identified gives results on par with the mathematical model in statistical terms. So, the best algorithm out of four algorithm proposed in this paper is proved to be superior to all other algorithms for all sizes of problems and its performance is equal to that of the mathematical model for small size and medium size problems.

Algorithms for a network design problem with crossing supermodular demands

AbstractWe present approximation algorithms for a class of directed network design problems. The network design problem is to find a minimum cost subgraph such that for each vertex set S there are at least f(S) arcs leaving the set S. In the last 10 years general techniques have been developed for designing approximation algorithms for undirected network design problems. Recently, Kamal Jain gave a 2‐approximation algorithm for the case when the function f is weakly supermodular. There has been very little progress made on directed network design problems. The main techniques used for the undirected problems do not have simple extensions to the directed case. András Frank has shown that in a special case when the function f is intersecting supermodular the problem can be solved optimally. In this article, we use this result to get a 2‐approximation algorithm for a more general case when f is crossing supermodular. We also extend Jain's techniques to directed problems. We prove that if the function f is crossing supermodular, then any basic solution of the LP relaxation of our problem contains at least one variable with value greater or equal to ¼. This result implies a 4‐approximation algorithm for the class of directed network design problems. © 2004 Wiley Periodicals, Inc.

A primal–dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs

Recently, Becker and Geiger and Bafna, Berman and Fujito gave 2-approximation algorithms for the feedback vertex set problem in undirected graphs. We show how their algorithms can be explained in terms of the primal–dual method for approximation algorithms, which has been used to derive approximation algorithms for network design problems. In the process, we give a new integer programming formulation for the feedback vertex set problem whose integrality gap is at worst a factor of two; the well-known cycle formulation has an integrality gap of Θ( log n) , as shown by Even, Naor, Schieber and Zosin. We also give a new 2-approximation algorithm for the problem which is a simplification of the Bafna et al. algorithm.

Approximation Algorithms for Network Design Problems on Bounded Subsets

We address the problem of designing a network so that certain connectivity requirements are satisfied, at minimum cost of the edges used. The requirements are specified for each subset of vertices in terms of the number of edges with one endpoint in the set. We address a class of such problems, where the connectivity is required for sets of bounded size,p. We show that for fixedp(p≤2), the problem is NP-complete, regardless of whether the requirement function is proper or not. For this class of bounded network design problems, we describe anO(logp)-approximation algorithm for the case of a proper requirement function. We introduce a relaxation of the network design problem ondirected graphs, where the requirement of each set is satisfied by the arcs that have their tails in the set. The relaxation allows arcswithinthe set to count toward the requirement of the set. We show that this relaxed problem is polynomial-time solvable if the requirement function isproper. The corresponding relaxation of the undirected problem is also polynomial-time solvable when the requirement function is proper. It is shown that in the case where the requirement function is not proper, the relaxed version is NP-complete. The integer relaxed problem or its linear programming relaxation is used to produce an approximate solution to the network design problem. Our analysis illustrates that the requirement of properness affects the difficulty of the problem.