Algorithms for NP-hard Optimization Problems and Cluster Analysis

Algorithms for NP-hard Optimization Problems and Cluster Analysis by Nan Li The set cover problem, weighted set cover problem, minimum dominating set problem and minimum weighted dominating set problem are all classical NP-hard optimization problems of great importance in both theory and real applications. Since the exact algorithms, which require exhaustive exploration of exponentially many options, are infeasible in practice, approximation algorithms and heuristic algorithms are widely used to find reasonably good solutions in polynomial time. I propose novel algorithms for these four problems. My algorithms for the weighted set cover and minimum weighted dominating set problems are based on a three-step strategy. For the weighted set cover problem, in the first step, we reserve the sets indispensable for the optimal solution and reduce the problem size. In the second step, we build a robust solution with a novel greedy heuristic. Sets are iteratively selected according to a measure which integrates the weight, the coverage gain for the current iteration and the global coverage capacity of each set. It favors the sets that have smaller weights and better extend or consolidate the coverage, especially on the items that are contained in less sets. Since the obtained solution tends to have a robust coverage, in the third step, we further improve it by removing the redundant sets in an efficient way. For the minimum weighted dominating set problem, we first reserve the indispensable vertices for the optimal solution. Then we convert it into a weighted set cover problem to solve it. These two algorithms can be used to solve the set cover problem and minimum dominating set problem by simply considering all the sets or vertices as having the same weights. Extensive experimental evaluations on a large number of synthetic and real-world set cover instances and graphs from many domains demonstrate the