Approximation Algorithms for Independent Set Problems on Hypergraphs

This thesis deals with approximation algorithms for the Maximum Independent Set and the Minimum Hitting Set problems on hypergraphs. As a hypergraph is a generalization of a graph, the question is whether the best known approximations on graphs can be extended to hypergraphs. We consider greedy, local search and partitioning algorithms. We introduce a general technique, called shrinkage reduction, that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to prove approximation ratios for greedy and local search algorithms for the Maximum Weak Independent Set problem on weighted and unweighted bounded-degree hypergraphs. For the weighted case we improve bounds using a simple partitioning algorithm. We also consider two variations of the max-greedy algorithms for the Maximum Strong Independent Set problem. We describe an SDP-based approach for the Maximum Weak Independent Set problem on bounded-degree hypergraphs. Our approach is to use semidefinite technique to sparsify a given hypergraph and then apply combinatorial algorithms to find a large independent set in the resulting sparser instance. Using this approach we obtain the first performance ratio that is sublinear in terms of the maximum or average degree of the hypergraph. We extend this result to the weighted case and give a similar bound in terms of the average weighted degree in a hypergraph, matching the best bounds known for the special case of graphs. We present several randomized and deterministic algorithms for the Maximum Weak Independent Set problem in a semi-streaming model. All our semi-streaming algorithms require only one pass over the stream and most of them resemble on-line algorithms in maintaining a feasible solution at all times. We introduce the on-line minimal space semi-streaming model and give lower and upper bounds for deterministic and randomized algorithms in this model. Nálgunarreiknirit fyrir óháð mengi í ofurnetum