Berge's conjecture on directed path partitions—a survey

AbstractBerge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k⩾1. The conjecture is still open and intriguing for all k>1.11Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k=2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians

Revisiting path-type covering and partitioning problems

This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts

Parameterizing Path Partitions

We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains \NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results also transfer to the case of covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of the CIAC 2023 conferenc

Asset Pricing and Trading Volume

Ce mémoire de thèse est organisé en trois articles. Le premier est dédié au cas des moulins de Toulouse dont les données nous permettent de tester certains points de la théorie de l’évaluation des actifs. Plus précisément, nous proposons une mesure de la consommation locale et réalisons une analyse basée sur l’entropie relative pour extraire le facteur stochastique d’actualisation de cette économie. Nous observons que ce dernier est lié à la consommation et qu’un modèle simple à la Lucas n’est pas rejeté pour des niveaux d’aversion pour le risque bas. Dans le second article, nous décrivons de manière purement théorique la relation entre le volume d’échange et la composition du marché par le biais d’un modèle où les préférences d’un agent dépendent de son environnement et où un choc de liquidité peut survenir de manière collective pour tous les membres d’un même groupe. Nous introduisons alors le concept de canal désirable comme condition nécessaire à la réalisation d’un échange et lions la topologie du réseau au volume espéré des échanges. Le troisième article porte sur le rôle des statuts sociaux dans la dynamique de marché. Nous proposons un modèle où deux types de biens sont disponibles, un bien positionnel et un bien non positionnel. En distinguant dans l’économie ceux possédant un statut et ceux qui n’en possèdent pas nous justifions comment les échanges prennent place au cours du temps par rapport à cette distinction sociale. Les prédictions du modèle sont alors testées sur les données historiques des moulins de Toulouse.This doctoral thesis is organized in three articles. In the first one, we use the Toulouse mills companies data as a suitable testbed for asset pricing theory. More precisely, we provide a proxy for local consumption and perform a relative entropy analysis to extract the stochastic discount factor of this old economy. We found that the model-free pricing kernel correlates with consumption and a standard CRRA-model is not rejected by the data, even for very low risk aversion levels. In the second article, we describe the relationship between trading volume and market composition through a pure theoretical approach. We build a model where the agent preferences depend on his environment and a liquidity shock is collectively experienced by the members of each social group in the economy. We introduce the concept of desirability channel as a necessary condition for a trade to occur and we rely the topology of the network to the expected volume. The third article focus on the role of social status concern in the exchanges dynamic. We propose a setting where two types of goods are available, a positional and a non positional one. By splitting the economy into two social groups, we depict how trades take place over time regarding to these social groups. The model predictions are finally tested on the historical support of the Toulouse mills companies