All solution graphs in multidimensional screening

We study general discrete-types multidimensional screening without any noticeable restrictions on valuations, using instead epsilon-relaxation of the incentive-compatibility constraints. Any active (becoming equality) constraint can be perceived as "envy" arc from one type to another, so the set of active constraints is a digraph. We find that: (1) any solution has an in-rooted acyclic graph ("river"); (2) for any logically feasible river there exists a screening problem resulting in such river. Using these results, any solution is characterized both through its spanning-tree and through its Lagrange multipliers, that can help in finding solutions and their efficiency/distortion properties.incentive compatibility; multidimensional screening; second-degree price discrimination; non-linear pricing; graphs

The birth of the strong components

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + \mu n^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as $\mu$ goes from $-\infty$ to $\infty$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of $\mu$ and provide more properties of the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle-point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter $p$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2-cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.Comment: 62 pages, 12 figures, 6 tables. Supplementary computer algebra computations available at https://gitlab.com/vit.north/strong-components-au

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph

All solution graphs in multidimensional screening

We study general discrete-types multidimensional screening without any noticeable restrictions on valuations, using instead epsilon-relaxation of the incentive-compatibility constraints. Any active (becoming equality) constraint can be perceived as "envy" arc from one type to another, so the set of active constraints is a digraph. We find that: (1) any solution has an in-rooted acyclic graph ("river"); (2) for any logically feasible river there exists a screening problem resulting in such river. Using these results, any solution is characterized both through its spanning-tree and through its Lagrange multipliers, that can help in finding solutions and their efficiency/distortion properties

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Threshold phenomena involving the connected components of random graphs and digraphs

We consider some models of random graphs and directed graphs and investigate their behavior near thresholds for the appearance of certain types of connected components. Firstly, we look at the critical window for the appearance of a giant strongly connected component in binomial random digraphs. We provide bounds on the probability that the largest strongly connected component is very large or very small. Next, we study the configuration model for graphs and show new upper bounds on the size of the largest connected component in the subcritical and barely subcritical regimes. We also show that these bounds are tight in some instances. Finally we look at the configuration model for random digraphs. We investigate the barely sub-critical region and show that this model behaves similarly to the binomial random digraph whose barely sub- and super-critical behaviour was studied by Luczak and Seierstad. Moreover, we show the existence of a threshold for the existence of a giant weak component, as predicted by Kryven.En aquesta tesi considerem diversos models de grafs i graf dirigits aleatoris, i investiguem el seu comportament a prop dels llindars per l'aparició de certs tipus de components connexes. En primer lloc, estudiem la finestra crítica per a l'aparició d'una component fortament connexa en dígrafs aleatoris binomials (o d'Erdos-Rényi). En particular, provem diversos resultats sobre la probabilitat límit que la component fortament connexa sigui sigui molt gran o molt petita. A continuació, estudiem el model de configuració per a grafs no dirigits i mostrem noves cotes superiors per la mida de la component connexa més gran en els règims sub-crítics i quasi-subcrítics. També demostrem que, en general, aquestes cotes no poden ser millorades. Finalment, estudiem el model de configuració per a dígrafs aleatoris. Ens centrem en la regió quasi-subcrítica i demostrem que aquest model es comporta de manera similar al model binomial, el comportament del qual va ser estudiat per Luczak i Seierstad en les regions quasi-subcrítica i quasi-supercrítica. A més a més, demostrem l'existència d'una funció llindar per a l'existència d'una component feble gegant, tal com va predir Kryven.Postprint (published version

On optimal and near-optimal algorithms for some computational graph problems

PhD ThesisSome computational graph problems are considered in this thesis and algorithms for solving these problems are described in detail. The problems can be divided into three main classes, namely, problems involving partially ordered sets, finding cycles in graphs, and shortest path problems. Most of the algorithms are based on recursive procedures using depth-first search. The efficiency of each algorithm is derived and it can be concluded that the majority of the proposed algorithms are either optimal and near-optimal within a constant factor. The efficiency of the algorithms is measured by the time and space requirements for their implementation.Conselho Nacional de Pesquisas,Brazil: Universidade Federal do Rio de Janeiro, Brazil

All solution graphs in multidimensional screening

We study general discrete-types multidimensional screening without any noticeable restrictions on valuations, using instead epsilon-relaxation of the incentive-compatibility constraints. Any active (becoming equality) constraint can be perceived as "envy" arc from one type to another, so the set of active constraints is a digraph. We find that: (1) any solution has an in-rooted acyclic graph ("river"); (2) for any logically feasible river there exists a screening problem resulting in such river. Using these results, any solution is characterized both through its spanning-tree and through its Lagrange multipliers, that can help in finding solutions and their efficiency/distortion properties

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session