The generalized minimum spanning tree problem

We consider the Generalized Minimum Spanning Tree Problem denoted by GMSTP. It is known that GMSTP is NP-hard and even finding a near optimal solution is NP-hard. We introduce a new mixed integer programming formulation of the problem which contains a polynomial number of constraints and a polynomial number of variables. Based on this formulation we give an heuristic solution, a lower bound procedure and an upper bound procedure and present the advantages of our approach in comparison with an earlier method. We present a solution procedure for solving GMST problem using cutting planes

FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin

Transversal numbers over subsets of linear spaces

Let $M$ be a subset of $\mathbb{R}^k$. It is an important question in the theory of linear inequalities to estimate the minimal number $h=h(M)$ such that every system of linear inequalities which is infeasible over $M$ has a subsystem of at most $h$ inequalities which is already infeasible over $M.$ This number $h(M)$ is said to be the Helly number of $M.$ In view of Helly's theorem, $h(\mathbb{R}^n)=n+1$ and, by the theorem due to Doignon, Bell and Scarf, $h(\mathbb{Z}^d)=2^d.$ We give a common extension of these equalities showing that $h(\mathbb{R}^n \times \mathbb{Z}^d) = (n+1) 2^d.$ We show that the fractional Helly number of the space $M \subseteq \mathbb{R}^d$ (with the convexity structure induced by $\mathbb{R}^d$) is at most $d+1$ as long as $h(M)$ is finite. Finally we give estimates for the Radon number of mixed integer spaces

Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint.We relate the secondstage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.