On Optimal $w$-gons in Convex Polygons

TL;DR

This paper improves an algorithm for finding optimal convex $w$-gons within a set of points, reducing the time complexity from $O(wn^3)$ to $O(n^3 \log w)$, enhancing efficiency for convex polygons.

Contribution

The authors present an improved algorithm for the MDF problem, reducing its time complexity and optimizing the process of selecting convex $w$-gons from point sets.

Findings

Reduced the algorithm's complexity from $O(wn^3)$ to $O(n^3 \log w)$

Provided a more efficient method for convex $w$-gon optimization

Enhanced performance for convex polygon input cases

Abstract

Let $P$ be a set of $n$ points in $\mathbb{R}^2$. For a given positive integer $w<n$, our objective is to find a set $C \subset P$ of points, such that $CH(P\setminus C)$ has the smallest number of vertices and $C$ has at most $n-w$ points. We discuss the $O(wn^3)$ time dynamic programming algorithm for monotone decomposable functions (MDF) introduced for finding a class of optimal convex $w$-gons, with vertices chosen from $P$, and improve it to $O(n^3 \log w)$ time, which gives an improvement to the existing algorithm for MDFs if their input is a convex polygon.