General 2-path Problem

TL;DR

This paper investigates the conditions under which a general 2-path problem can be solved in linear time, providing necessary and sufficient criteria and analyzing the computational complexity of the associated algorithms.

Contribution

It introduces a comprehensive framework for characterizing the 2-path problem with new necessary and sufficient conditions, demonstrating that the problem is solvable in polynomial time.

Findings

Established necessary and sufficient conditions for the 2-path problem

Proved the problem is a P problem with polynomial-time algorithms

Analyzed the computational complexity of the solution algorithms

Abstract

In this paper, some preliminaries about signal flow graph, linear time-invariant system on F(z) and computational complexity are first introduced in detail. In order to synthesize the necessary and sufficient condition on F(z) for a general 2-path problem, the sufficient condition on F(z) or R and necessary conditions on F(z) for a general 2-path problem are secondly analyzed respectively. Moreover, an equivalent sufficient and necessary condition on R whether there exists a general 2-path is deduced in detail. Finally, the computational complexity of the algorithm for this equivalent sufficient and necessary condition is introduced so that it means that the general 2-path problem is a P problem.