arXiv subjects

Etienne de Klerk

Publications and source records attributed to Etienne de Klerk.

41 records · Page 3Linked to original sources

arXiv2014

An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r$). We show that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For general polynomials, if there exists a rational global minimizer over the simplex, we show that the convergence rate is also of the order $O(1/r^2)$. Our results answer a question posed by De Klerk et al. (2013) and improves on previously known $O(1/r)$ bounds in the quadratic case.

math.OC↗

arXiv2012

Book drawings of complete bipartite graphs

A "book" with k pages consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The pagenumber of a graph G is the minimum k such that G admits a k-page embedding (that is, a k-page drawing with no edge crossings). The k-page crossing number nu_k(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the pagenumbers and k-page crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact k-page crossing number of K_{k+1,n} for 3<=k<=6. We also prove the general asymptotic estimate lim_{k->oo} lim_{n->oo} nu_k(K_{k+1,n})/(2n^2/k^2)=1. Finally, we give general upper bounds for nu_k(K_{m,n}), and relate these bounds to the k-planar crossing numbers of K_{m,n} and K_n.

math.CO↗

arXiv2012

Improved lower bounds on book crossing numbers of complete graphs

A "book with k pages" consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number nu_k(G) of a graph G is the minimum number of crossings in a k-page drawing of G. In this paper we investigate the k-page crossing numbers of complete graphs K_n. We use semidefinite programming techniques to give improved lower bounds on nu_k(K_n) for various values of k. We also use a maximum satisfiability reformulation to calculate the exact value of nu_k(K_n) for several values of k and n. Finally, we investigate the best construction known for drawing K_n in k pages, calculate the resulting number of crossings, and discuss this upper bound in the light of the new results reported in this paper.

math.CO↗

arXiv2011

Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming

It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively, K_n) with exactly Z(m, n) (respectively, Z(n)) crossings, thus yielding the conjectures (I) v_2(Km,n) =Z(m,n), and (II) v_2(Kn) = Z(n). It is known that (I) holds for min{m, n} <=6, and that (II) holds for n<=14. In this paper we prove that (I) holds asymptotically (that is, lim_n v_2 (K_{m,n})/Z (m, n) = 1) for m=7 and 8. We also prove (II) for 15<=n<=18 and n=20,24, and establish the asymptotic estimate lim_n v_2(K_n)/Z(n) >= 0.9253. The previous best-known lower bound involved the constant 0.8594.

math.CO↗

arXiv2008

Bounding the Betti numbers and computing the Euler-Poincaré characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by \[ \ell^2 (O(s+\ell+m)\ell d)^{k+2m}. \] This is a common generalization of previous results on bounding the Betti numbers of closed semi-algebraic sets defined by polynomials of degree $d$ and 2, respectively. We also describe an algorithm for computing the Euler-Poincaré characteristic of such sets, generalizing similar algorithms known before. The complexity of the algorithm is bounded by $(\ell s m d)^{O(m(m+k))}$.

math.AG↗