Joseph Paat

Extreme Functions with an Arbitrary Number of Slopes

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number

The Structure of the Infinite Models in Integer Programming

k\ge 2

Knot mosaic tabulation

, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson that stated all extreme functions are piecewise linear.

Approximation of Corner Polyhedra with Families of Intersection Cuts

We contribute to the theory for minimal liftings of cut-generating functions. In particular, we give three operations that preserve the so-called covering property of certain structured cut-generating functions. This has the consequence of vastly expanding the set of undominated cut generating functions which can be used computationally, compared to known examples from the literature. The results of this paper are significant generalizations of previous results from the literature on such operations, and also use completely different proof techniques which we feel are more suitable for attacking future research questions in this area.

Extreme functions with an arbitrary number of slopes

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for finite dimensional corner polyhedra. One consequence is that nonnegative continuous functions suffice to describe finite dimensional corner polyhedra with rational data. We also discover new facts about corner polyhedra with non-rational data.

Conflict-Free Railway Track Assignment at Depots

Lomonaco and Kauffman [Quantum knots and mosaics, Quantum Inf. Process. 7(2–3) (2008) 85–115] introduced the notion of knot mosaics in their work on quantum knots. It is conjectured that knot mosaic type is a complete invariant of tame knots. In this paper, we answer a question of C. Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not. That is, there is an infinite family of non-minimal knots whose mosaic numbers are known.

Distances of optimal solutions of mixed-integer programs

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior of knot theory is assumed or required.

Distances between optimal solutions of mixed-integer programs

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor in fixed dimension n (the constant depends on n). The literature already contains several results in this direction. In this paper, we use the maximum number of facets of a lattice-free set in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that for each natural number n, a corner polyhedron for n integer variables is approximated by intersection cuts from lattice-free sets with at most i facets up to a constant factor (depending only on n) if \(i> 2^{n-1}\) and that no such approximation is possible if \(i \le 2^{n-1}\). When the approximation factor is allowed to depend on the denominator of the underlying fractional point of the corner polyhedron, we show that the threshold is \(i > n\) versus \(i \le n\). The tools introduced for proving such results are of independent interest for studying intersection cuts.