Uniform unweighted set cover: The power of non-oblivious local search

AbstractWe are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k (p<k). In addition, we assume that the input contains all possible subsets of size p. Our objective is to find a subcollection of minimum-cardinality which covers all the elements. This problem is known to be NP-hard. We provide two approximation algorithms for it, one for the generic case, and an improved one for the special case of (p,k)=(2,4).The algorithm for the generic case is a greedy one, based on packing phases: at each phase we pick a collection of disjoint subsets covering i new elements, starting from i=k down to i=p+1. At a final step we cover the remaining base elements by the subsets of size p. We derive the exact performance guarantee of this algorithm for all values of k and p, which is less than Hk, where Hk is the k’th harmonic number. However, the algorithm exhibits the known improvement methods over the greedy one for the unweighted k-set cover problem (in which subset sizes are only restricted not to exceed k), and hence it serves as a benchmark for our improved algorithm.The improved algorithm for the special case of (p,k)=(2,4) is based on non-oblivious local search: it starts with a feasible cover, and then repeatedly tries to replace sets of size 3 and 4 so as to maximize an objective function which prefers big sets over small ones. For this case, our generic algorithm achieves an asymptotic approximation ratio of 1.5+ϵ, and the local search algorithm achieves a better ratio, which is bounded by 1.458333…+ϵ

Survey of Approximation Algorithms for Set Cover Problem

In this thesis, I survey 11 approximation algorithms for unweighted set cover problem. I have also implemented the three algorithms and created a software library that stores the code I have written. The algorithms I survey are: 1. Johnson's standard greedy; 2. f-frequency greedy; 3. Goldsmidt, Hochbaum and Yu's modified greedy; 4. Halldorsson's local optimization; 5. Dur and Furer semi local optimization; 6. Asaf Levin's improvement to Dur and Furer; 7. Simple rounding; 8. Randomized rounding; 9. LP duality; 10. Primal-dual schema; and 11. Network flow technique. Most of the algorithms surveyed are refinements of standard greedy algorithm

Heuristic algorithms for wireless mesh network planning

x, 131 leaves : ill. ; 29 cmTechnologies like IEEE 802.16j wireless mesh networks are drawing increasing attention of the research community. Mesh networks are economically viable and may extend services such as Internet to remote locations. This thesis takes interest into a planning problem in IEEE 802.16j networks, where we need to establish minimum cost relay and base stations to cover the bandwidth demand of wireless clients. A special feature of this planning problem is that any node in this network can send data to at most one node towards the next hop, thus traffic flow is unsplittable from source to destination. We study different integer programming formulations of the problem. We propose four types of heuristic algorithms that uses greedy, local search, variable neighborhood search and Lagrangian relaxation based approaches for the problem. We evaluate the algorithms on database of network instances of 500-5000 nodes, some of which are randomly generated network instances, while other network instances are generated over geometric distribution. Our experiments show that the proposed algorithms produce satisfactory result compared to benchmarks produced by generalized optimization problem solver software

Proceedings of the 8th international conference on disability, virtual reality and associated technologies (ICDVRAT 2010)

The proceedings of the conferenc