Spanning simple path inside a simple polygon

Given a set S of n colored points of m colors inside a simple polygon P, each point within the polygon has a specific color that is not necessarily unique, i.e., they may exhibit the same color. The study aims to find a simple path that traverses at least one point of each color using a set of S points contained within a simple polygon P. Two results are presented in this study. First, we demonstrate that finding such simple paths inside a simple polygon is an NP-complete\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NP-complete$$\end{document} problem. Moreover, we provide a polynomial-time algorithm that computes the simple path when P is an orthogonal spiral simple polygon, and our objective is to locate a simple Hamiltonian path L using all points of S inside P. Our algorithm has a time complexity of O(r+rn4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(r+rn^4)$$\end{document}, where r is the number of reflex vertices in P and n is the number of points in S.