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

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

Effective computability of solutions of ordinary differential equations: the thousand monkeys approach

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz

A Colorful Steinitz Lemma with Applications to Block Integer Programs

The Steinitz constant in dimension $d$ is the smallest value $c(d)$ such that for any norm on $\mathbb{R}^{ d}$ and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by $c(d)$. Grinberg and Sevastyanov prove that $c(d) \le d$ and that the bound of $d$ is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences. Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.Comment: Shortened proofs, fixed typos, and streamlined the argument in Section

Effective computability of solutions of differential inclusions-the ten thousand monkeys approach

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

The Bernstein basis in set-theoretic geometric modelling

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN037062 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Vectors in a box

For an integer [various formulas omitted]. The quantity t(d) was introduced by Dash, Fukasawa, and Günlük, who showed that [various formulas omitted]. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of [various formulas omitted]. 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 t(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