arXiv subjects
Publications and source records attributed to Basile Couëtoux.
In 2012, Nešetřil and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $Ω(\log \log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, 2-degenerate graphs that have a path of order $n$ and where the longest induced paths have order $O((\log \log n)^{1+o(1)})$.
We introduce splitter networks, which abstract the behavior of conveyor belts found in the video game Factorio. Based on this definition, we show how to compute the steady-state of a splitter network. Then, leveraging insights from the players community, we provide multiple designs of splitter networks capable of load-balancing among several conveyor belts, and prove that any load-balancing network on $n$ belts must have $Ω(n \log n)$ nodes. Incidentally, we establish connections between splitter networks and various concepts including flow algorithms, flows with equality constraints, Markov chains and the Knuth-Yao theorem about sampling over rational distributions using a fair coin.
In this paper we consider the problem of optimizing the ecological connectivity of a landscape under a budget constraint by improving habitat areas and ecological corridors between them. We consider a formulation of this problem in terms of graphs in which vertices represent the habitat areas and arcs represent a probability of connection between two areas that depend on the quality of the respective corridor. We propose a new generalized flow model that improves existing models for this problem and an efficient preprocessing algorithm that reduces the size of the graphs on which generalized flows is computed. Reported numerical experiments highlight the benefice of these two contributions on computation times and show that larger problems can be solved using them. Our experiments also show that several variants of greedy algorithms perform relatively well on practical instances while they return arbitrary bad solutions in the worst case.