A faster algorithm for packing branchings in digraphs

Lee, Orlando; Leston-Rey, Mario

doi:10.1016/j.dam.2015.05.016

Cited by 2 publications

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“…Também existem resultados recentes sobre redes temporais [KK15], onde a direção dos arcos correspondem a uma modelagem desse tipo de rede. Devido a sua importância, o problema de arborescência, e algumas generalizações relacionadas, tem sido bastante estudado em versões algorítmicas como pode ser visto em [GM98,Kam14,LRW15,LLR15] Por outro lado, quando trabalhamos com empacotamento de grafos não geradores estamos procurando empacotar subgrafos que têm a quantidade de vértices menor do que os grafos onde procuramos empacotar. Como exemplo, o problema clássico do emparelhamento máximo pode ser visto como o empacotamento de arestas disjuntas nos vértices.…”

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Empacotamento e contagem em digrafos: cenários aleatórios e extremais

Parente¹

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In this thesis we study two problems dealing with digraphs: a packing problem and a counting problem. We study the problem of packing the maximum number of arborescences in the random digraph D(n, p), where each possible arc is included uniformly at random with probability p = p(n). Let λ(D(n, p)) denote the largest integer λ ≥ 0 such that, for all 0 ≤ ≤ λ, we have −1 i=0 (− i)|{v : d in (v) = i}| ≤. We show that the maximum number of arc-disjoint arborescences in D(n, p) is λ(D(n, p)) asymptotically almost surely. We also give tight estimates for λ(D(n, p)) for every p ∈ [0, 1]. The main tools that we used were expansion properties of random digraphs, the behavior of in-degree of random digraphs and a classic result by Frank relating subpartitions and number of arborescences. For the counting problem, we study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct of this study we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5. We use the theory of flag algebras as a main tool for this problem and combinatorial settings obtained from semidefinite method.

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