A Fixed Parameter Algorithm for the Minimum Number Convex Partition Problem

Abstract

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixed-parameter tractable algorithm for this problem that runs in the following time complexities:

– linear time if k is constant,

– time polynomial in n if $k=O(\frac{{\rm log}n}{{\rm log log}n})$,

or, to be exact, in O(n k $^{\rm 6{\it k}-5}$ 2^16k) time.