On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

Approximate min-max relations on plane graphs

Let G be a plane graph, let τ(G) (resp. τ′(G)) be the minimum number of vertices (resp. edges) that meet all cycles of G, and let ν(G) (resp. ν′(G)) be the maximum number of vertex-disjoint (resp. edge-disjoint) cycles in G. In this note we show that τ(G)≤3 ν(G) and τ′(G)≤4 ν′(G)-1; our proofs are constructive, which yield polynomial-time algorithms for finding corresponding objects with the desired properties. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201

Half-integral Erd\H{o}s-P\'osa property of directed odd cycles

We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral Erd\H{o}s-P\'osa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.Comment: 16 pages, 5 figure

Approximate min-max relations for odd cycles in planar graphs

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Diâmetro de grafos fulerenes e transversalidade de ciclos ímpares de fuleróides-(3, 4, 5, 6)

Fullerene graphs are mathematical models for molecules composed exclusively of carbon atoms, discovered experimentally in the early 1980s by Kroto, Heath, O’Brien, Curl and Smalley. Many parameters associated to these graphs have been discussed, trying to describe the stability of the fullerene’s molecule. Formally, fulerene graphs are 3-connected, cubic, planar graphs with pentagonal and hexagonal faces. Andova and Škrekovski Conjecture [1] states that the diameter of all fullerene graph, on n vertices, is at least equal to jq 5n 3 k −1. This conjecture became relevant, since Andova and Škrekovski conceived it from the study of highly regular, spherical and symmetrical fullerene graphs. We introduce the concepts of combinatorial curvature of vertex and combinatorial curvature of face of a planar graph and then we define a specific class of fullerene graphs, called fullerene nanodiscs. We have shown that the Andova and Škrekovski Conjecture is not valid for any fullerene nanodisc with more than 300 vertices. However, we exhibit infinite classes of fullerene graphs, similar to the graphs studied by Andova and Škrekovski, which satisfy this conjecture. Adding to fullerene graphs, triangular and quadrangular faces we conceive fuleroid-(3, 4, 5, 6) graphs. We studied the bipartite edge frustration and the maximum independent set problems on the fuleroid-(3, 4, 5, 6) graphs, obtaining tight limits for both problems.Os grafos fulerenes são modelos matemáticos para moléculas compostas exclusivamente por átomos de carbono, descobertas experimentalmente no início da década de 80 por Kroto, Heath, O’Brien, Curl e Smalley. Muitos parâmetros associados a estes grafos vêm sendo discutidos, buscando descrever a estabilidade das moléculas de fulerene. Precisamente falando, grafos fulerenes são grafos cúbicos, planares, 3-conexos cujas faces são pentagonais e hexagonais. A Conjectura de Andova e Škrekovski [1] afirma que o diâmetro de todo grafo fulerene, contendo n vértices, é pelo menos igual a jq 5n 3 k − 1. Esta conjectura tornou-se relevante, pois Andova e Škrekovski conceberam-na a partir do estudo de grafos fulerenes altamente regulares, esféricos e simétricos. Introduzimos os conceitos de curvatura combinatória de vértice e curvatura combinatória de face de um grafo planar. Definimos, então, uma classe particular de grafos fulerenes, chamada de nanodiscos de fulerene. Mostramos que a Conjectura de Andova e Škrekovski não é válida para nenhum nanodisco de fulerene com mais de 300 vértices. No entanto, exibimos infinitas classes de grafos fulerenes, semelhantes aos grafos estudados por Andova e Škrekovski, que satisfazem a referida conjectura. Acrescentando, aos grafos fulerenes, faces triangulares e quadrangulares concebemos os grafos fuleróides-(3, 4, 5, 6). Estudamos os problemas da frustração bipartida de arestas e do conjunto independente máximo sobre os grafos fuleróides-(3, 4, 5, 6), obtendo limites apertados para ambos os problemas