ABSTRACT
Randomization is one of the most interesting tools in designing efficient algorithms. A remarkable property of randomized algorithms is their structural simplicity. In fact, in several cases, although the known deterministic algorithms are quite involved, the randomized ones are simpler and much easier to code. NP-hard optimization problems are not known to allow finding optimal solutions efficiently. Their combinatorial structure is elaborate and sometimes quite cryptic in the general sense. The set cover problem is one of the oldest known NP-complete problems. The Markov chain Monte Carlo (MCMC) method is a development of the classic, well-known Monte Carlo method for approximately estimating measures and quantities whose exact computation is a difficult task. Proving satisfactory upper bounds for the mixing time of the simulated Markov chain is in fact the most interesting (nontrivial) point in the application of the MCMC method.
