Sequential Optimization Numbers and Conjecture about Edge-Symmetry and Weight-Symmetry Shortest Weight-Constrained Path (2206.07052v1)

Abstract: This paper defines multidimensional sequential optimization numbers and prove that the unsigned Stirling numbers of first kind are 1-dimensional sequential optimization numbers. This paper gives a recurrence formula and an upper bound of multidimensional sequential optimization numbers. We proof that the k-dimensional sequential optimization numbers, denoted by O_k (n,m), are almost in {O_k (n,a)}, where a belong to[1,eklog(n-1)+(epi)^2/6(2^k-1)+M_1], n is the size of k-dimensional sequential optimization numbers and M_1 is large positive integer. Many achievements of the Stirling numbers of first kind can be transformed into the properties of k-dimensional sequential optimization numbers by k-dimensional extension and we give some examples. Shortest weight-constrained path is NP-complete problem [1]. In the case of edge symmetry and weight symmetry, we use the definition of the optimization set to design 2-dimensional BeLLMan-Ford algorithm to solve it. According to the fact that P_1 (n,m>M) less than or equal to e^-M_1 , where M=elog(n-1)+e+M_1, M_1 is a positive integer and P_1 (n,m) is the probability of 1-dimensional sequential optimization numbers, this paper conjecture that the probability of solving edge-symmetry and weight-symmetry shortest weight-constrained path problem in polynomial time approaches 1 exponentially with the increase of constant term in algorithm complexity. The results of a large number of simulation experiments agree with this conjecture.