Solution of a Generalized Problem of Covering by means of a Generic Algorithm

Natela Ananiashvili
2017 Baltic Journal of Modern Computing  
An algorithm of solution of a generalized problem of covering is offered. Generalization is made on the basis of subsets with diverse values of minimal summary weight of the given set. In difference from a standard problem of covering, this problem implies selection of particular covering, when rank sum of covering columns (from the covering matrix) gives the particular vector and corresponding values of this vector exceeds not a singular vector, but pre-existing vector b (with nonsingular
more » ... nents). Offered algorithm is based on a general genetic algorithm. It uses variable operator of non-standard mutation. One of the first and the best of approximate algorithm is offered by Chvatal (Chvatal, 1979) . It solves the standard problem of covering in polynomial time. The work of Grossman and Wool (1997) considers the standard problem of covering, offers a heuristic algorithm and argues that it gives better results than previous techniques of solution. Lan et al. (2007) present meta-heuristic algorithm of standard covering. Ananiashvili (2015a) gives precise solutions of problems of the least division and covering and uses technique of branches and limits. According to this work, by means of compact insertion ("packaging") of columns of a covering matrix, a volume of random access memory (used for calculations) and number of operations is decreased approximately 32 times. Ananiashvili (2015b) offers approximate solution of the standard problem of covering by means of modified genetic algorithm and represents results of test problems. In their paper, Azar et al (2009) considered the general problem and gave a logarithmic approximation algorithm for it. In their paper, Bansal et al. (2010) improved their result and gave a simple randomized constant factor approximation algorithm for the generalized min-sum set cover problem. Lim et al. (2014) offer greedy algorithm of solution of problem of minimal covering. Yang and Leung (2003) and Umetani et al (2013) consider the generalized problem of weighted covering and require multiple covering of every element. For comparison, the authors use commercial software CPLEX (ILOG COMPLEX 7.0 -User's Manual, ILOG) which is used for solution of problems of integer programming and they use it for their own test problems. Motivation is that they don't know results of other authors. Yesipov and Muraviev (2014) offer two algorithms: additive and genetic. The efficiency of these algorithms is tested on randomly developed matrices and vector of weights is also filled randomly.
doi:10.22364/bjmc.2017.5.4.02 fatcat:qeqqjolm6zaizkm45g3igpw2z4