PERMANENT DECOMPOSITION ALGORITHM FOR THE COMBINATORIAL OBJECTS GENERATION

Abstract

Context. The problem of generating vectors consisting of different representatives of a given set of sets is considered. Such problems arise, in particular, in scheduling theory, when scheduling appointments. A special case of this problem is the problem of generating permutations. Objective. Problem is considered from the point of view of a permanent approach and a well-known one, based on the concept of lexicographic order. Method. In many tasks, it becomes necessary to generate various combinatorial objects: permutations, combinations with and without repetitions, various subsets. In this paper we consider a new approach to the combinatorial objects generation, which is based on the procedure of the permanent decomposition. Permanent is built for the special matrix of incidence. The main idea of this approach is including to the process of the algebraic permanent decomposition by row additional function for the column identifiers writing into corresponding data structures. In this case, the algebraic permanent in not calculated, but we get a specific recursive algorithm for generating a combinatorial object. The computational complexity of this algorithm is analyzed. Results. It is investigated a new approach to the generation of complex combinatorial objects, based on the procedure of decomposition of the modified permanent of the incidence matrix by line with memorization of index elements. Conclusions. The permanent algorithms of the combinatorial objects generation is investigated. The complexity of our approach in the case of permutation is compared with the lexicographic algorithm and the Johnson-Trotter algorithm. The obtained results showed that our algorithm belongs to the same complexity class as the lexicographic algorithm and the Johnson-Trotter method. Numerical results confirmed the effectiveness of our approach.