Vectors in a Box

For an integer d>=1, let tau(d) be the smallest integer with the following property: If v1,v2,...,vt is a sequence of t>=2 vectors in [-1,1]^d with v1+v2+...+vt in [-1,1]^d, then there is a subset S of {1,2,...,t} of indices, 2<=|S|<=tau(d), such that \sum_{i\in S} vi is in [-1,1]^d. The quantity tau(d) was introduced by Dash, Fukasawa, and G\"unl\"uk, who showed that tau(2)=2, tau(3)=4, and tau(d)=Omega(2^d), and asked whether tau(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of tau(d) <= d^{d+o(d)}, and based on a construction of Alon and Vu, whose main idea goes back to Hastad, we obtain a lower bound of tau(d)>= d^{d/2-o(d)}. These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al., which is a "universal" polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on tau(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.Comment: 12 pages, 1 figur

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

A Quantitative Steinitz Theorem for Plane Triangulations

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3 \times 8n^5 \times \zeta(n)$ integer grid, where $\zeta(n) \leq (500 n^8)^{\tau(G)}$ and $\tau(G)$ denotes the shedding diameter of $G$, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure

Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

We consider integer programming problems in standard form $\max \{c^Tx : Ax = b, \, x\geq 0, \, x \in Z^n\}$ where $A \in Z^{m \times n}$, $b \in Z^m$ and $c \in Z^n$. We show that such an integer program can be solved in time $(m \Delta)^{O(m)} \cdot \|b\|_\infty^2$, where $\Delta$ is an upper bound on each absolute value of an entry in $A$. This improves upon the longstanding best bound of Papadimitriou (1981) of $(m\cdot \Delta)^{O(m^2)}$, where in addition, the absolute values of the entries of $b$ also need to be bounded by $\Delta$. Our result relies on a lemma of Steinitz that states that a set of vectors in $R^m$ that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by $m$. We also use the Steinitz lemma to show that the $\ell_1$-distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by $m \cdot (2\,m \cdot \Delta+1)^m$. Here $\Delta$ is again an upper bound on the absolute values of the entries of $A$. The novel strength of our bound is that it is independent of $n$. We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.Comment: We achieve much milder dependence of the running time on the largest entry in $b

Many projectively unique polytopes

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in S^d, a new Alexandrov--van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat