The recent paper A Quantitative Doignon-Bell-Scarf Theorem by Aliev et al. [ Combinatorica, 37 (2017), pp. 313--332] generalizes the famous Doignon--Bell--Scarf theorem on the existence of integer solutions to systems of linear inequalities. Their generalization examines the number of facets of a polyhedron that contains exactly $k$ integer points in ${R}^n$. They show that there exists a number $c(n,k)$ such that any polyhedron in $\mathbb{R}^n$ that contains exactly $k$ integer points has a relaxation to at most $c(n,k)$ of its inequalities that will define a new polyhedron with the same integer points. They prove that $c(n,k) = O(k)2^n$. In this paper, we improve the bound asymptotically to be sublinear in $k$, that is, $c(n,k) = o(k) 2^n$. We also provide lower bounds on $c(n,k)$, along with other structural results. For dimension n=2, our upper and lower bounds match to within a constant factor.

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on n vertices, with a polynomial number of constraints, requires many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first nontrivial lower bound on the number of integer variables needed in such settings.

We present a generic approach that allows us to develop a fully polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions over the integer points in a rational polyhedron in fixed dimension. The approach combines the subdivision strategy of Papadimitriou and Yannakakis [22] with ideas similar to those commonly used to derive real algebraic certificates of positivity for polynomials. Our general approach is widely applicable. We apply it, for instance, to the Motzkin polynomial and to indefinite quadratic forms xT Qx in a fixed number of variables, where Q has at most one positive, or at most one negative eigenvalue. In dimension three, this leads to an FPTAS for general Q.