Complexity and Stop Conditions for NP as General Assignment Problems, the Travel Salesman Problem in $\mathbb{R}^2$, Knight Tour Problem and Boolean Satisfiability Problem

TL;DR

This paper introduces specific stop conditions for solving various NP problems, including TSP, Knight Tour, and SAT, based on geometric and topological properties, with algorithms and examples demonstrating their application.

Contribution

It proposes novel stop conditions for NP problems grounded in geometric and topological criteria, enhancing solution strategies for these problems.

Findings

Jordan's simple curve condition for TSP trajectories

Crossing condition for Knight Tour feasibility

Application of these conditions to SAT problem analysis

Abstract

This paper presents stop conditions for solving General Assignment Problems (GAP), in particular for Travel Salesman Problem in an Euclidian 2D space the well known condition Jordan's simple curve and opposite condition for the Knight Tour Problem. The Jordan's simple curve condition means that a optimal trajectory must be simple curve, i.e., without crossing but for Knight Tour Problem we use the contrary, the feasible trajectory must have crossing in all cities of the tour. The paper presents the algorithms, examples and some results come from Concorde's Home page. Several problem are studied to depict their properties. A classical decision problem SAT is studied in detail.