LIPIcs

The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs

Lower bounds on dynamic programming for maximum weight independent set

Publisher Copyright: © 2021 Tuukka Korhonen.We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2Ω(w/d) gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2Ω(w) gates. An MWIS-formula is an MWIScircuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2Ω(t/d) gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.Peer reviewe

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $\Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le \Delta^\frac{2\varepsilon}{1+2\varepsilon}/(\log\Delta)^2$, with $\lfloor(1+\varepsilon)\Delta/\log(\Delta/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.Comment: 23 page

Algorithms for Fundamental Problems in Computer Networks.

Traditional studies of algorithms consider the sequential setting, where the whole input data is fed into a single device that computes the solution. Today, the network, such as the Internet, contains of a vast amount of information. The overhead of aggregating all the information into a single device is too expensive, so a distributed approach to solve the problem is often preferable. In this thesis, we aim to develop efficient algorithms for the following fundamental graph problems that arise in networks, in both sequential and distributed settings. Graph coloring is a basic symmetry breaking problem in distributed computing. Each node is to be assigned a color such that adjacent nodes are assigned different colors. Both the efficiency and the quality of coloring are important measures of an algorithm. One of our main contributions is providing tools for obtaining colorings of good quality whose existence are non-trivial. We also consider other optimization problems in the distributed setting. For example, we investigate efficient methods for identifying the connectivity as well as the bottleneck edges in a distributed network. Our approximation algorithm is almost-tight in the sense that the running time matches the known lower bound up to a poly-logarithmic factor. For another example, we model how the task allocation can be done in ant colonies, when the ants may have different capabilities in doing different tasks. The matching problems are one of the classic combinatorial optimization problems. We study the weighted matching problems in the sequential setting. We give a new scaling algorithm for finding the maximum weight perfect matching in general graphs, which improves the long-standing Gabow-Tarjan's algorithm (1991) and matches the running time of the best weighted bipartite perfect matching algorithm (Gabow and Tarjan, 1989). Furthermore, for the maximum weight matching problem in bipartite graphs, we give a faster scaling algorithm whose running time is faster than Gabow and Tarjan's weighted bipartite {it perfect} matching algorithm.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113540/1/hsinhao_1.pd

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Coloring problems in combinatorics and descriptive set theory

In this dissertation we study problems related to colorings of combinatorial structures both in the “classical” finite context and in the framework of descriptive set theory, with applications to topological dynamics and ergodic theory. This work consists of two parts, each of which is in turn split into a number of chapters. Although the individual chapters are largely independent from each other (with the exception of Chapters 4 and 6, which partially rely on some of the results obtained in Chapter 3), certain common themes feature throughout—most prominently, the use of probabilistic techniques. In Chapter 1, we establish a generalization of the Lovász Local Lemma (a powerful tool in probabilistic combinatorics), which we call the Local Cut Lemma, and apply it to a variety of problems in graph coloring. In Chapter 2, we study DP-coloring (also known as correspondence coloring)—an extension of list coloring that was recently introduced by Dvorák and Postle. The goal of that chapter is to gain some understanding of the similarities and the differences between DP-coloring and list coloring, and we find many instances of both. In Chapter 3, we adapt the Lovász Local Lemma for the needs of descriptive set theory and use it to establish new bounds on measurable chromatic numbers of graphs induced by group actions. In Chapter 4, we study shift actions of countable groups on spaces of the form A, where A is a finite set, and apply the Lovász Local Lemma to find “large” closed shift-invariant subsets X A on which the induced action of is free. In Chapter 5, we establish precise connections between certain problems in graph theory and in descriptive set theory. As a corollary of our general result, we obtain new upper bounds on Baire measurable chromatic numbers from known results in finite combinatorics. Finally, in Chapter 6, we consider the notions of weak containment and weak equivalence of probability measure-preserving actions of a countable group—relations introduced by Kechris that are combinatorial in spirit and involve the way the action interacts with finite colorings of the underlying probability space. This work is based on the following papers and preprints: [Ber16a; Ber16b; Ber16c; Ber17a; Ber17b; Ber17c; Ber18a; Ber18b], [BK16; BK17a] (with Alexandr Kostochka), [BKP17] (with Alexandr Kostochka and Sergei Pron), and [BKZ17; BKZ18] (with Alexandr Kostochka and Xuding Zhu)

On algorithms for large-scale graph and clustering problems

Gegenstand dieser Arbeit sind algorithmische Methoden der modernen Datenanalyse. Dabei werden vorwiegend zwei übergeordnete Themen behandelt: Datenstromalgorithmen mit Kompressionseigenschaften und Approximationsalgorithmen für Clusteringverfahren. Datenstromalgorithmen verarbeiten einen Datensatz sequentiell und haben das Ziel, Eigenschaften des Datensatzes (approximativ) zu bestimmen, ohne dabei den gesamten Datensatz abzuspeichern. Unter Clustering versteht man die Partitionierung eines Datensatzes in verschiedene Gruppen. Das erste dargestellte Problem betrifft Matching in Graphen. Hier besteht der Datensatz aus einer Folge von Einfüge- und Löschoperationen von Kanten. Die Aufgabe besteht darin, die Größe des so genannten Maximum Matchings so genau wie möglich zu bestimmen. Es wird ein Algorithmus vorgestellt, der, unter der Annahme, dass das Matching höchstens die Größe k hat, die exakte Größe bestimmt und dabei k² Speichereinheiten benötigt. Dieser Algorithmus lässt sich weiterhin verwenden um eine konstante Approximation der Matchinggröße in planaren Graphen zu bestimmen. Des Weiteren werden untere Schranken für den benötigten Speicherplatz bestimmt und eine Reduktion von gewichtetem Matching zu ungewichteten Matching durchgeführt. Anschließend werden Datenstromalgorithmen für die Nachbarschaftssuche betrachtet, wobei die Aufgabe darin besteht, für n gegebene Mengen die Paare mit hoher Ähnlichkeit in nahezu Linearzeit zu finden. Dabei ist der Jaccard Index |A ∩ B|/|A U B| das Ähnlichkeitsmaß für zwei Mengen A und B. In der Arbeit wird eine Datenstruktur beschrieben, die dies erstmalig in dynamischen Datenströmen mit geringem Speicherplatzverbrauch leistet. Dabei werden Zufallszahlen mit nur 2-facher Unabhängigkeit verwendet, was eine sehr effiziente Implementierung ermöglicht. Das dritte Problem befindet sich an der Schnittstelle zwischen den beiden Themen dieser Arbeit und betrifft das k-center Clustering Problem in Datenströmen mit einem Zeitfenster. Die Aufgabe besteht darin k Zentren zu finden, sodass die maximale Distanz unter allen Punkten zu dem jeweils nächsten Zentrum minimiert wird. Ergebnis sind ein 6-Approximationalgorithmus für ein beliebiges k und ein optimaler 4-Approximationsalgorithmus für k = 2. Die entwickelten Techniken lassen sich ebenfalls auf das Durchmesserproblem anwenden und ermöglichen für dieses Problem einen optimalen Algorithmus. Danach werden Clusteringprobleme bezüglich der Jaccard Distanz analysiert. Dabei sind wieder eine Menge N von Teilmengen aus einer Grundgesamtheit U sind und die Aufgabe besteht darin eine Teilmenge $C$ zu finden, die max 1-|X ∩ C|/|X U C| minimiert. Es wird gezeigt, dass zwar eine exakte Lösung des Problems NP-schwer ist, es aber gleichzeitig eine PTAS gibt. Abschließend wird die weit verbreitete lokale Suchheuristik für k-median und k-means Clustering untersucht. Obwohl es im Allgemeinen schwer ist, diese Probleme exakt oder auch nur approximativ zu lösen, gelten sie in der Praxis als relativ gut handhabbar, was andeutet, dass die Härteresultate auf pathologischen Eingaben beruhen. Auf Grund dieser Diskrepanz gab es in der Vergangenheit praxisrelevante Datensätze zu charakterisieren. Für drei der wichtigsten Charakterisierungen wird das Verhalten einer lokalen Suchheuristik untersucht mit dem Ergebnis, dass die lokale Suchheuristik in diesen Fällen optimale oder fast optimale Cluster ermittelt