A convex partition of a point set P in the plane is a planar partition of the convex hull of P into empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition with the minimum number of internal faces. The problem has been shown to be NP-hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.

The color coding technique is used to solve subgraph isomorphism problems, in particular path problems. One color among C is randomly assigned to each vertex of the graph and if distinct colors are given to the vertices of the desired subgraph, it can be found efficiently by dynamic programming. These two phases are repeated until the subgraph is found with a high probability, which can require a large number of iterations. We propose new coloring strategies that take advantage of the graph structure to increase this probability and thus reduce the number of iterations. They provide a guaranteed improvement over the original color coding technique based on a particular structural parameter related to the bandwidth. When this parameter is smaller than the number C of colors, we prove that only C calls to the dynamic program are needed to find the subgraph.

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, no two terminals on the same horizontal or vertical line, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) belong to the axis-parallel grid defined by T and are oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this article, we present a polynomial factor 2-approximation algorithm for the bidirected minimum Manhattan network problem. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(2), 167–178 2017