Integer decomposition for polyhedra defined by nearly totally unimodular matrices

We call a matrix $A$ nearly totally unimodular if it can be obtained from a totally unimodular matrix $\tilde{A}$ by adding to each row of $\tilde{A}$ an integer multiple of some fixed row a^{\ssf T} of $\tilde{A}$. For an integer vector $b$ and a nearly totally unimodular matrix $A$, we denote by $P_{A,b}$ the integer hull of the set ${x\in{\Bbb R}^n\mid Ax\leq b}$. We show that $P_{A,b}$ has the integer decomposition property and that we can find a decomposition of a given integer vector $x\in kP_{A,b}$ in polynomial time

Semidefinite code bounds based on quadruple distances

Let $A(n,d)$ be the maximum number of $0,1$ words of length $n$, any two having Hamming distance at least $d$. We prove $A(20,8)=256$, which implies that the quadruply shortened Golay code is optimal. Moreover, we show $A(18,6)\leq 673$, $A(19,6)\leq 1237$, $A(20,6)\leq 2279$, $A(23,6)\leq 13674$, $A(19,8)\leq 135$, $A(25,8)\leq 5421$, $A(26,8)\leq 9275$, $A(21,10)\leq 47$, $A(22,10)\leq 84$, $A(24,10)\leq 268$, $A(25,10)\leq 466$, $A(26,10)\leq 836$, $A(27,10)\leq 1585$, $A(25,12)\leq 55$, and $A(26,12)\leq 96$. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for $A(n,d)$. The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of $n$ and $d$.Comment: 15 page

A combinatorial identity arising from cobordism theory

Let $\underline\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)\in{\Bbb R}_{>0}^m$. Let $\mathop{\underline\alpha\,}_{i,j}$ be the vector obtained from $\underline\alpha$ by deleting the entries $\alpha_i$ and $\alpha_j$. A. Besser and P. Moree [Arch. Math. (Basel) 79 (2002), no. 6, 463--471; MR1967264 (2004a:11014)] introduced some invariants and near invariants related to the solutions \underline\epsilon\in\{\pm1}^{m-2} of the linear inequality ${|\alpha_i-\alpha_j|}<\langle\underline\epsilon, \mathop{\underline\alpha\,}_{i,j}\rangle<\alpha_i+\alpha_j$, where $\langle·,·\rangle$ denotes the usual inner product and $\mathop{\underline\alpha\,}_{i,j}$ the vector obtained from $\underline\alpha$ by deleting $\alpha_i$ and $\alpha_j$. The main result of [op. cit.] is extended here to a much more general setting, namely that of certain maps from finite sets to ${-1,1}$

New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming

We give a new upper bound on the maximum size $A_q(n,d)$ of a code of word length $n$ and minimum Hamming distance at least $d$ over the alphabet of $q\geq 3$ letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in $n$ using semidefinite programming. For $q=3,4,5$ this gives several improved upper bounds for concrete values of $n$ and $d$. This work builds upon previous results of A. Schrijver [IEEE Trans. Inform. Theory 51 (2005), no. 8, 2859--2866] on the Terwilliger algebra of the binary Hamming schem

An algorithm for weighted fractional matroid matching

Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find for any given weights on the lines in L, a maximum weight fractional matching

Wiskunde, wat moet je er mee? RTL4's Editie NL, 17.07.2012 [3:03]

Een filmploeg van RTL heeft opnames gemaakt op het CWI voor het RTL-programma Editie NL. Aanleiding is de gouden medaillewinst van twee Nederlandse scholieren in de Internationale Wiskunde Olympiade. Met o.a. interviews met Rob van der Mei en Dion Gijswijt, en beelden van de Alan Turingtentoonstelling