Finding cores of random 2-SAT formulae via Poisson cloning

For the random 2-SAT formula $F(n,p)$, let $F_C (n,p)$ be the formula left after the pure literal algorithm applied to $F(n,p)$ stops. Using the recently developed Poisson cloning model together with the cut-off line algorithm (COLA), we completely analyze the structure of $F_{C} (n,p)$. In particular, it is shown that, for \gl:= p(2n-1) = 1+\gs with \gs\gg n^{-1/3}, the core of $F(n,p)$ has \thl^2 n +O((\thl n)^{1/2}) variables and \thl^2 \gl n+O((\thl n))^{1/2} clauses, with high probability, where \thl is the larger solution of the equation \th- (1-e^{-\thl \gl})=0. We also estimate the probability of $F(n,p)$ being satisfiable to obtain \pr[ F_2(n, \sfrac{\gl}{2n-1}) is satisfiable ] = \caseth{1-\frac{1+o(1)}{16\gs^3 n}}{if $\gl= 1-\gs$ with $\gs\gg n^{-1/3}$}{}{}{e^{-\Theta(\gs^3n)}}{if $\gl=1+\gs$ with $\gs\gg n^{-1/3}$,} where $o(1)$ goes to 0 as \gs goes to 0. This improves the bounds of Bollob\'as et al. \cite{BBCKW}

The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k >= 3. The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity.Comment: This is the long version of a paper that will be published in the SIAM Journal on Discrete Mathematics. This long version includes an appendix and a computer program. The contents of the paper are unchanged in the latest version. The format of the arxiv submission was changed so that the computer program will appear as an ancillary file. Some comments in the computer program were update

Core Structures in Random Graphs and Hypergraphs

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges

Random hypergraphs for hashing-based data structures

This thesis concerns dictionaries and related data structures that rely on providing several random possibilities for storing each key. Imagine information on a set S of m = |S| keys should be stored in n memory locations, indexed by [n] = {1,…,n}. Each object x [ELEMENT OF] S is assigned a small set e(x) [SUBSET OF OR EQUAL TO] [n] of locations by a random hash function, independent of other objects. Information on x must then be stored in the locations from e(x) only. It is possible that too many objects compete for the same locations, in particular if the load c = m/n is high. Successfully storing all information may then be impossible. For most distributions of e(x), however, success or failure can be predicted very reliably, since the success probability is close to 1 for loads c less than a certain load threshold c^* and close to 0 for loads greater than this load threshold. We mainly consider two types of data structures: • A cuckoo hash table is a dictionary data structure where each key x [ELEMENT OF] S is stored together with an associated value f(x) in one of the memory locations with an index from e(x). The distribution of e(x) is controlled by the hashing scheme. We analyse three known hashing schemes, and determine their exact load thresholds. The schemes are unaligned blocks, double hashing and a scheme for dynamically growing key sets. • A retrieval data structure also stores a value f(x) for each x [ELEMENT OF] S. This time, the values stored in the memory locations from e(x) must satisfy a linear equation that characterises the value f(x). The resulting data structure is extremely compact, but unusual. It cannot answer questions of the form “is y [ELEMENT OF] S?”. Given a key y it returns a value z. If y [ELEMENT OF] S, then z = f(y) is guaranteed, otherwise z may be an arbitrary value. We consider two new hashing schemes, where the elements of e(x) are contained in one or two contiguous blocks. This yields good access times on a word RAM and high cache efficiency. An important question is whether these types of data structures can be constructed in linear time. The success probability of a natural linear time greedy algorithm exhibits, once again, threshold behaviour with respect to the load c. We identify a hashing scheme that leads to a particularly high threshold value in this regard. In the mathematical model, the memory locations [n] correspond to vertices, and the sets e(x) for x [ELEMENT OF] S correspond to hyperedges. Three properties of the resulting hypergraphs turn out to be important: peelability, solvability and orientability. Therefore, large parts of this thesis examine how hyperedge distribution and load affects the probabilities with which these properties hold and derive corresponding thresholds. Translated back into the world of data structures, we achieve low access times, high memory efficiency and low construction times. We complement and support the theoretical results by experiments.Diese Arbeit behandelt Wörterbücher und verwandte Datenstrukturen, die darauf aufbauen, mehrere zufällige Möglichkeiten zur Speicherung jedes Schlüssels vorzusehen. Man stelle sich vor, Information über eine Menge S von m = |S| Schlüsseln soll in n Speicherplätzen abgelegt werden, die durch [n] = {1,…,n} indiziert sind. Jeder Schlüssel x [ELEMENT OF] S bekommt eine kleine Menge e(x) [SUBSET OF OR EQUAL TO] [n] von Speicherplätzen durch eine zufällige Hashfunktion unabhängig von anderen Schlüsseln zugewiesen. Die Information über x darf nun ausschließlich in den Plätzen aus e(x) untergebracht werden. Es kann hierbei passieren, dass zu viele Schlüssel um dieselben Speicherplätze konkurrieren, insbesondere bei hoher Auslastung c = m/n. Eine erfolgreiche Speicherung der Gesamtinformation ist dann eventuell unmöglich. Für die meisten Verteilungen von e(x) lässt sich Erfolg oder Misserfolg allerdings sehr zuverlässig vorhersagen, da für Auslastung c unterhalb eines gewissen Auslastungsschwellwertes c* die Erfolgswahrscheinlichkeit nahezu 1 ist und für c jenseits dieses Auslastungsschwellwertes nahezu 0 ist. Hauptsächlich werden wir zwei Arten von Datenstrukturen betrachten: • Eine Kuckucks-Hashtabelle ist eine Wörterbuchdatenstruktur, bei der jeder Schlüssel x [ELEMENT OF] S zusammen mit einem assoziierten Wert f(x) in einem der Speicherplätze mit Index aus e(x) gespeichert wird. Die Verteilung von e(x) wird hierbei vom Hashing-Schema festgelegt. Wir analysieren drei bekannte Hashing-Schemata und bestimmen erstmals deren exakte Auslastungsschwellwerte im obigen Sinne. Die Schemata sind unausgerichtete Blöcke, Doppel-Hashing sowie ein Schema für dynamisch wachsenden Schlüsselmengen. • Auch eine Retrieval-Datenstruktur speichert einen Wert f(x) für alle x [ELEMENT OF] S. Diesmal sollen die Werte in den Speicherplätzen aus e(x) eine lineare Gleichung erfüllen, die den Wert f(x) charakterisiert. Die entstehende Datenstruktur ist extrem platzsparend, aber ungewöhnlich: Sie ist ungeeignet um Fragen der Form „ist y [ELEMENT OF] S?“ zu beantworten. Bei Anfrage eines Schlüssels y wird ein Ergebnis z zurückgegeben. Falls y [ELEMENT OF] S ist, so ist z = f(y) garantiert, andernfalls darf z ein beliebiger Wert sein. Wir betrachten zwei neue Hashing-Schemata, bei denen die Elemente von e(x) in einem oder in zwei zusammenhängenden Blöcken liegen. So werden gute Zugriffszeiten auf Word-RAMs und eine hohe Cache-Effizienz erzielt. Eine wichtige Frage ist, ob Datenstrukturen obiger Art in Linearzeit konstruiert werden können. Die Erfolgswahrscheinlichkeit eines naheliegenden Greedy-Algorithmus weist abermals ein Schwellwertverhalten in Bezug auf die Auslastung c auf. Wir identifizieren ein Hashing-Schema, das diesbezüglich einen besonders hohen Schwellwert mit sich bringt. In der mathematischen Modellierung werden die Speicherpositionen [n] als Knoten und die Mengen e(x) für x [ELEMENT OF] S als Hyperkanten aufgefasst. Drei Eigenschaften der entstehenden Hypergraphen stellen sich dann als zentral heraus: Schälbarkeit, Lösbarkeit und Orientierbarkeit. Weite Teile dieser Arbeit beschäftigen sich daher mit den Wahrscheinlichkeiten für das Vorliegen dieser Eigenschaften abhängig von Hashing Schema und Auslastung, sowie mit entsprechenden Schwellwerten. Eine Rückübersetzung der Ergebnisse liefert dann Datenstrukturen mit geringen Anfragezeiten, hoher Speichereffizienz und geringen Konstruktionszeiten. Die theoretischen Überlegungen werden dabei durch experimentelle Ergebnisse ergänzt und gestützt

Fundamental Approaches to Software Engineering

This open access book constitutes the proceedings of the 24th International Conference on Fundamental Approaches to Software Engineering, FASE 2021, which took place during March 27–April 1, 2021, and was held as part of the Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg but changed to an online format due to the COVID-19 pandemic. The 16 full papers presented in this volume were carefully reviewed and selected from 52 submissions. The book also contains 4 Test-Comp contributions