Matchings, coverings, and Castelnuovo-Mumford regularity

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

A Quantum Lovasz Local Lemma

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of $\Omega(2^k/k^2)$. Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.Comment: 19 page

On covering by translates of a set

In this paper we study the minimal number of translates of an arbitrary subset $S$ of a group $G$ needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, we show that while the worst-case efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and $n$ large, almost every $k$-subset of any given $n$-element group covers $G$ with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and Algorithm

Fractional coverings, greedy coverings, and rectifier networks

A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i,j) iff there is a path from the j-th source to the i-th sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century. We explore two techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area. First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpinski (disjointness) matrix. The previous upper bound is O(n^{1.28}), and we improve it to O(n^{1.17}), while the best known lower bound is Omega(n^{1.16}). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)

Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings

This thesis consists of three independent parts. The first part of the thesis is concerned with Ramsey theory. Given an integer $q\geq 2$, a graph $G$ is said to be \emph{$q$-Ramsey} for another graph $H$ if in any $q$-edge-coloring of $G$ there exists a monochromatic copy of $H$. The central line of research in this area investigates the smallest number of vertices in a $q$-Ramsey graph for a given $H$. In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) $q$-Ramsey graph for a given $H$. This line of research was initiated by Burr, Erdős, and Lovász in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given $H$ can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph $H$ affect the corresponding collection of $q$-Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and Szabó and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph $K_t$ changes the collection of 2-Ramsey graphs significantly. The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order $n$} is an $n\times n$ array with entries in $[n]$ such that each integer appears exactly once in every row and every column. Two Latin squares $L$ and $L'$ are said to be {\em orthogonal} if, for all $x,y\in [n]$, there is a unique pair $(i,j)\in [n]^2$ such that $L(i,j) = x$ and $L'(i,j) = y$; a system of {\em $k$ mutually orthogonal Latin squares}, or a {\em $k$-MOLS}, is a set of $k$ pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order $n$ log-asymptotically, we study the number of $k$-MOLS of order $n$. Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of $k = k(n)$. We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to. The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least $k$ times, while covering the remaining point fewer times. We study this problem for the grid $\mathbb{F}_2^n$, determining the number exactly when one of the parameters $n$ and $k$ is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for $k=1$. Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube $\set{0,1}^n\subseteq\mathbb{R}^n$, we study hyperplane coverings for different grids over $\mathbb{R}$, under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen. Der erste Teil beschäftigt sich mit Ramseytheorie. Für eine ganze Zahl $q\geq 2$ nennt man einen Graphen \emph{$q$-Ramsey} f\"ur einen anderen Graphen $H$, wenn jede Kantenf\"arbung mit $q$ Farben einen einfarbigen Teilgraphen enthält, der isomorph zu $H$ ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschäftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezüglich Teilgraphen) $q$-Ramsey-Graphen f\"ur einen gegebenen Graphen $H$. Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes $H$ enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller $q$-Ramsey-Graphen f\"ur $H$ verändert, wenn wir den Graphen $H$ modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und Szabó und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus $K_t$ mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu $K_t$. Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung $n$} ist eine $n\times n$-Matrix, gef\"ullt mit den Zahlen aus $[n]$, in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle $x,y\in[n]$ genau ein Paar $(i,j)\in [n]^2$ existiert, sodass es $L(i,j) = x$ und $L'(i,j) = y$ gilt. Ein \emph{k-MOLS der Ordnung $n$} ist eine Menge von $k$ lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung $n$ log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele $k$-MOLS der Ordnung $n$ es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern $k=k(n)$. Zusätzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann. Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu überdecken, sodass ein bestimmter Punkt maximal $(k-1)$-mal bedeckt ist und alle andere mindestens $k$-mal. Wir untersuchen diese Anzahl f\"ur das Gitter $\mathbb{F}_2^n$ asymptotisch und sogar genau, wenn eins von $n$ und $k$ viel größer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison für den Fall $k=1$. Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt überhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den Hyperwürfel $\set{0,1}^n\subseteq \mathbb{R}^n$ untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können

Subspace coverings with multiplicities

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k-1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and F\"uredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k \ge 2^{n-d-1}$ we have $f(n,k,d)=2^d k - \left \lfloor \frac{k}{2^{n-d}} \right \rfloor$, while for $n > 2^{2^d k-k-d+1}$ we have $f(n,k,d)= n + 2^dk-d-2$, and also study the transition between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.Comment: 15 page

When Does a Mixture of Products Contain a Product of Mixtures?

We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table

Sequential legislative lobbying

In this paper, we analyze the equilibrium of a sequential game-theoretical model of lobbying, due to Groseclose and Snyder (1996), describing a legislature that vote over two alternatives, where two opposing lobbies, Lobby 0 and Lobby 1, compete by bidding for legislators’ votes. In this model, the lobbyist moving first suffers from a second mover advantage and will make an offer to a panel of legislators only if it deters any credible counter-reaction from his opponent, i.e., if he anticipates to win the battle. This paper departs from the existing literature in assuming that legislators care about the consequence of their votes rather than their votes per se. Our main focus is on the calculation of the smallest budget that he needs to win the game and on the distribution of this budget across the legislators. We study the impact of the key parameters of the game on these two variables and show the connection of this problem with the combinatorics of sets and notions from cooperative game theory.Lobbying; cooperative games; noncooperative games