Intersecting and 22-intersecting hypergraphs with maximal covering number: the Erd\H{o}s-Lov\'asz theme revisited

Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges in an intersecting $r$-uniform hypergraph. It was known that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, $q(5)=13$, and we found 3 examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erd\H{o}s and Lov\'asz, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for $r\in \{3,4,5,6\}$. For uniformity 6, there is a unique extremal example. In a related question, we try to find $2$-intersecting $r$-uniform hypergraphs with maximal covering number, that is $\tau(H)=r-1$. An infinite family of examples is to take all possible $r$-sets of a $(2r-2)$-vertex set. There is also a geometric candidate: biplanes. These are symmetric 2-designs with $\lambda=2$. We determined that only 3 biplanes of the 18 known examples are extremal

Intersecting and 2-intersecting hypergraphs with maximal covering number: The Erdős–Lovász theme revisited

Erdős and Lovász noticed that an (Formula presented.) -uniform intersecting hypergraph (Formula presented.) with maximal covering number, that is, (Formula presented.), must have at least (Formula presented.) edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let (Formula presented.) denote the minimum number of edges in an intersecting (Formula presented.) -uniform hypergraph. It was known that (Formula presented.) and (Formula presented.). We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, (Formula presented.), and we found three examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erdős and Lovász, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for (Formula presented.). For uniformity 6, there is a unique extremal example. In a related question, we try to find 2-intersecting r-uniform hypergraphs with maximal covering number, that is, (Formula presented.). An infinite family of examples is to take all possible r-sets of a (Formula presented.) -vertex set. There is also a geometric candidate: biplanes. These are symmetric 2-designs with (Formula presented.). We determined that only three biplanes of the 18 known examples are extremal. © 2020 Wiley Periodicals LL

Viewing extremal and structural problems through a probabilistic lens

This thesis focuses on using techniques from probability to solve problems from extremal and structural combinatorics. The main problem in Chapter 2 is determining the typical structure of $t$-intersecting families in various settings and enumerating such systems. The analogous sparse random versions of our extremal results are also obtained. The proofs follow the same general framework, in each case using a version of the Bollobás Set-Pairs Inequality to bound the number of maximal intersecting families, which then can be combined with known stability theorems. Following this framework from joint work with Balogh, Das, Liu, and Sharifzadeh, similar results for permutations, uniform hypergraphs, and vector spaces are obtained. In 2006, Barát and Thomassen conjectured that the edges of every planar 4-edge-connected 4-regular graph can be decomposed into disjoint copies of $S_3$, the star with three leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Following joint work with Postle, in Chapter 3 using the Small Subgraph Conditioning Method of Robinson and Wormald, we find that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided we have the obvious necessary divisibility conditions. In 1988, Thomassen showed that if $G$ is at least $(2k-1)$-edge-connected then $G$ has a spanning, bipartite $k$-connected subgraph. In 1989, Thomassen asked whether a similar phenomenon holds for vertex-connectivity; more precisely: is there an integer-valued function $f(k)$ such that every $f(k)$-connected graph admits a spanning, bipartite $k$-connected subgraph? In Chapter 4, as in joint work with Ferber, we show that every $10^{10}k^3 \log n$-connected graph admits a spanning, bipartite $k$-connected subgraph

Efficient approximations of conjunctive queries

When finding exact answers to a query over a large database is infeasible, it is natural to approximate the query by a more efficient one that comes from a class with good bounds on the complexity of query evaluation. In this paper we study such approximations for conjunctive queries. These queries are of special importance in databases, and we have a very good understanding of the classes that admit fast query evaluation, such as acyclic, or bounded (hyper)treewidth queries. We define approximations of a given query Q as queries from one of those classes that disagree with Q as little as possible. We mostly concentrate on approximations that are guaranteed to return correct answers. We prove that for the above classes of tractable conjunctive queries, approximations always exist, and are at most polynomial in the size of the original query. This follows from general results we establish that relate closure properties of classes of conjunctive queries to the existence of approximations. We also show that in many cases, the size of approximations is bounded by the size of the query they approximate. We establish a number of results showing how combinatorial properties of queries affect properties of their approximations, study bounds on the number of approximations, as well as the complexity of finding and identifying approximations. We also look at approximations that return all correct answers and study their properties

Matchings and Flows in Hypergraphs

In this dissertation, we study matchings and flows in hypergraphs using combinatorial methods. These two problems are among the best studied in the field of combinatorial optimization. As hypergraphs are a very general concept, not many results on graphs can be generalized to arbitrary hypergraphs. Therefore, we consider special classes of hypergraphs, which admit more structure, to transfer results from graph theory to hypergraph theory. In Chapter 2, we investigate the perfect matching problem on different classes of hypergraphs generalizing bipartite graphs. First, we give a polynomial time approximation algorithm for the maximum weight matching problem on so-called partitioned hypergraphs, whose approximation factor is best possible up to a constant. Afterwards, we look at the theorems of König and Hall and their relation. Our main result is a condition for the existence of perfect matchings in normal hypergraphs that generalizes Hall’s condition for bipartite graphs. In Chapter 3, we consider perfect f-matchings, f-factors, and (g,f)-matchings. We prove conditions for the existence of (g,f)-matchings in unimodular hypergraphs, perfect f-matchings in uniform Mengerian hypergraphs, and f-factors in uniform balanced hypergraphs. In addition, we give an overview about the complexity of the (g,f)-matching problem on different classes of hypergraphs generalizing bipartite graphs. In Chapter 4, we study the structure of hypergraphs that admit a perfect matching. We show that these hypergraphs can be decomposed along special cuts. For graphs it is known that the resulting decomposition is unique, which does not hold for hypergraphs in general. However, we prove the uniqueness of this decomposition (up to parallel hyperedges) for uniform hypergraphs. In Chapter 5, we investigate flows on directed hypergraphs, where we focus on graph-based directed hypergraphs, which means that every hyperarc is the union of a set of pairwise disjoint ordinary arcs. We define a residual network, which can be used to decide whether a given flow is optimal or not. Our main result in this chapter is an algorithm that computes a minimum cost flow on a graph-based directed hypergraph. This algorithm is a generalization of the network simplex algorithm.Diese Arbeit untersucht Matchings und Flüsse in Hypergraphen mit Hilfe kombinatorischer Methoden. In Graphen gehören diese Probleme zu den grundlegendsten der kombinatorischen Optimierung. Viele Resultate lassen sich nicht von Graphen auf Hypergraphen verallgemeinern, da Hypergraphen ein sehr abstraktes Konzept bilden. Daher schauen wir uns bestimmte Klassen von Hypergraphen an, die mehr Struktur besitzen, und nutzen diese aus um Resultate aus der Graphentheorie zu übertragen. In Kapitel 2 betrachten wir das perfekte Matchingproblem auf Klassen von „bipartiten“ Hypergraphen, wobei es verschiedene Möglichkeiten gibt den Begriff „bipartit“ auf Hypergraphen zu definieren. Für sogenannte partitionierte Hypergraphen geben wir einen polynomiellen Approximationsalgorithmus an, dessen Gütegarantie bis auf eine Konstante bestmöglich ist. Danach betrachten wir die Sätze von Konig und Hall und untersuchen deren Zusammenhang. Unser Hauptresultat ist eine Bedingung für die Existenz von perfekten Matchings auf normalen Hypergraphen, die Halls Bedingung für bipartite Graphen verallgemeinert. Als Verallgemeinerung von perfekten Matchings betrachten wir in Kapitel 3 perfekte f-Matchings, f-Faktoren und (g, f)-Matchings. Wir beweisen Bedingungen für die Existenz von (g, f)-Matchings auf unimodularen Hypergraphen, perfekten f-Matchings auf uniformen Mengerschen Hypergraphen und f-Faktoren auf uniformen balancierten Hypergraphen. Außerdem geben wir eine Übersicht über die Komplexität des (g, f)-Matchingproblems auf verschiedenen Klassen von Hypergraphen an, die bipartite Graphen verallgemeinern. In Kapitel 4 untersuchen wir die Struktur von Hypergraphen, die ein perfektes Matching besitzen. Wir zeigen, dass diese Hypergraphen entlang spezieller Schnitte zerlegt werden können. Für Graphen weiß man, dass die so erhaltene Zerlegung eindeutig ist, was im Allgemeinen für Hypergraphen nicht zutrifft. Wenn man jedoch uniforme Hypergraphen betrachtet, dann liefert jede Zerlegung die gleichen unzerlegbaren Hypergraphen bis auf parallele Hyperkanten. Kapitel 5 beschäftigt sich mit Flüssen in gerichteten Hypergraphen, wobei wir Hypergraphen betrachten, die auf gerichteten Graphen basieren. Das bedeutet, dass eine Hyperkante die Vereinigung einer Menge von disjunkten Kanten ist. Wir definieren ein Residualnetzwerk, mit dessen Hilfe man entscheiden kann, ob ein gegebener Fluss optimal ist. Unser Hauptresultat in diesem Kapitel ist ein Algorithmus, um einen Fluss minimaler Kosten zu finden, der den Netzwerksimplex verallgemeinert