Sequential Optimization Numbers and Conjecture about Edge-Symmetry and Weight-Symmetry Shortest Weight-Constrained Path

TL;DR

This paper introduces multidimensional sequential optimization numbers, explores their properties, and conjectures that certain symmetric shortest path problems can be solved in polynomial time with high probability as complexity increases.

Contribution

It defines and analyzes multidimensional sequential optimization numbers, relates them to Stirling numbers, and proposes a conjecture on polynomial-time solvability of symmetric shortest path problems.

Findings

Unsigned Stirling numbers of first kind are 1-dimensional sequential optimization numbers.

Provides recurrence formula and upper bounds for multidimensional sequential optimization numbers.

Conjectures exponential approach to polynomial-time solution probability for symmetric shortest path problems.

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.