Convex sets with lattice point constraints

Covering point sets with two convex objects

Let P2n be a point set in the plane with n red and n blue points. Let CR and CB (SR and SB) respectively be red and blue colored and disjoint disks (axisparallel squares). In this paper we prove the following results. Finding the positions for CR and CB that maximizes the number of red points covered by CR plus the number of blue points covered by CB can be done in O(n log n) time. Finding two ...

An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form S := {x : π0 ≤ π T x ≤ π0 + 1} called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be a polyhedron. A split set S has max-facet-width equal to one in the sense that max{π x : x ∈ S}− min{π...

On k-convex point sets

We extend the (recently introduced) notion of k-convexity of a two-dimensional subset of the Euclidean plane to finite point sets. A set of n points is ∗Corresponding author. Email addresses: [email protected] (Oswin Aichholzer), [email protected] (Franz Aurenhammer), [email protected] (Thomas Hackl), [email protected] (Ferran Hurtado), [email protected] (Alexander Pilz), pedro.ramos@uah...

Representations of lattice point sets

An algorithm reconstructing convex lattice sets

In this paper, we study the problem of reconstructing special lattice sets from X-rays in a finite set of prescribed directions. We present the class of “Q-convex” sets which is a new class of subsets of Z2 having a certain kind of weak connectedness. The main result of this paper is a polynomial-time algorithm solving the reconstruction problem for the “Q-convex” sets. These sets are uniquely ...