# Maximum rectilinear convex subsets

## TL;DR

This paper introduces efficient algorithms for finding subsets of points with maximum rectilinear convex properties, including boundary points, interior emptiness, and area, with applications to related polygon problems.

## Contribution

It presents new algorithms with $O(n^3)$ time and $O(n^2)$ space for various maximum rectilinear convex subset problems, improving understanding and methods in computational geometry.

## Key findings

- Algorithms for maximum boundary points in rectilinear convex hulls
- Algorithms for maximum area rectilinear convex hulls with empty interior
- Simplified algorithms for maximum-area orthoconvex and staircase polygons

## Abstract

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$, (2) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$ such that the rectilinear convex hull of $S$ has maximum area and its interior contains no element of $P$, and (4) when each point of $P$ is assigned a weight, positive or negative, a subset $S$ of $P$ that maximizes the total weight of the points in the rectilinear convex hull of $S$. We also revisit the problems of computing a maximum-area orthoconvex polygon and computing a maximum-area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.