Hardness and Approximation of Minimum Convex Partition

TL;DR

This paper proves the NP-hardness of the Minimum Convex Partition problem, introduces several approximation algorithms with different guarantees, and relates it to a covering problem, providing both hardness and algorithmic solutions.

Contribution

It establishes NP-hardness for the problem, develops multiple approximation algorithms, and connects the problem to a covering segments problem with an FPT algorithm.

Findings

NP-hardness of Minimum Convex Partition established

Approximation algorithms with O(log OPT) and O(√n log n) guarantees provided

Relation to Covering Points with Non-Crossing Segments enables FPT algorithms

Abstract

We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an O(log OPT)-approximation running in O(n^8)-time, where OPT denotes the minimum number of convex faces needed, to an O(sqrt(n) log n)-approximation algorithm running in O(n^2)-time. We say that a point set is k-directed if the (straight) lines containing at least three points have up to k directions. We present an O(k)-approximation algorithm running in n^O(k)-time. Those hardness and approximation results also holds for the Minimum Convex Tiling problem, defined similarly but allowing the use of Steiner points. The approximation results are obtained by relating the problem to the Covering Points with Non-Crossing Segments problem. We show that this problem is NP-hard, and present an FPT algorithm. This allows us to obtain a constant-approximation FPT algorithm for the Minimum Convex Partition Problem where the parameter is the number of faces.