# Geometric secluded paths and planar satisfiability

## TL;DR

This paper studies paths with minimal visibility in polygonal domains, providing approximation algorithms and complexity results, and explores connections to satisfiability problems.

## Contribution

It introduces a PTAS for minimum-exposure paths with integral exposure and analyzes complexity for 0/1 exposure, linking the problem to planar satisfiability.

## Key findings

- PTAS developed for integral exposure paths in polygons.
- Shortest path minimizes exposure in simple polygons, NP-hard with holes.
- Connections established between geometric exposure and planar satisfiability.

## Abstract

We consider paths with low \emph{exposure} to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between \emph{integral} exposure (when we care about how long the path sees every point of the domain) and \emph{0/1} exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT.