The Polygon Burning Problem

11/17/2021
by   William Evans, et al.
0

Motivated by the k-center problem in location analysis, we consider the polygon burning (PB) problem: Given a polygonal domain P with h holes and n vertices, find a set S of k vertices of P that minimizes the maximum geodesic distance from any point in P to its nearest vertex in S. Alternatively, viewing each vertex in S as a site to start a fire, the goal is to select S such that fires burning simultaneously and uniformly from S, restricted to P, consume P entirely as quickly as possible. We prove that PB is NP-hard when k is arbitrary. We show that the discrete k-center of the vertices of P under the geodesic metric on P provides a 2-approximation for PB, resulting in an O(n^2 log n + hkn log n)-time 3-approximation algorithm for PB. Lastly, we define and characterize a new type of polygon, the sliceable polygon. A sliceable polygon is a convex polygon that contains no Voronoi vertex from the Voronoi diagram of its vertices. We give a dynamic programming algorithm to solve PB exactly on a sliceable polygon in O(kn^2) time.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/19/2019

Hardness and approximation for the geodetic set problem in some graph classes

In this paper, we study the computational complexity of finding the geod...
research
03/09/2018

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram in a Simple Polygon

The geodesic Voronoi diagram of m point sites inside a simple polygon of...
research
10/25/2017

The Geodesic 2-center Problem in a Simple Polygon

The geodesic k-center problem in a simple polygon with n vertices consis...
research
01/13/2022

Approximate the individually fair k-center with outliers

In this paper, we propose and investigate the individually fair k-center...
research
02/05/2018

The Sea Exploration Problem: Data-driven Orienteering on a Continuous Surface

This paper describes a problem arising in sea exploration, where the aim...
research
03/17/2020

The Parameterized Complexity of Guarding Almost Convex Polygons

Art Gallery is a fundamental visibility problem in Computational Geometr...
research
11/27/2020

Minmax Regret 1-Sink Location Problems on Dynamic Flow Path Networks with Parametric Weights

This paper addresses the minmax regret 1-sink location problem on dynami...

Please sign up or login with your details

Forgot password? Click here to reset