12 research outputs found
From DNF Compression to Sunflower Theorems via Regularity
The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold (Computational Complexity, 2013). In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture
Improved bounds for the sunflower lemma
A sunflower with petals is a collection of sets so that the
intersection of each pair is equal to the intersection of all. Erd\H{o}s and
Rado proved the sunflower lemma: for any fixed , any family of sets of size
, with at least about sets, must contain a sunflower. The famous
sunflower conjecture is that the bound on the number of sets can be improved to
for some constant . In this paper, we improve the bound to about
. In fact, we prove the result for a robust notion of sunflowers,
for which the bound we obtain is tight up to lower order terms.Comment: Revised preprint, added sections on applications and rainbow
sunflower
Monotone circuit lower bounds from robust sunflowers
Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity Rossman (SIAM J. Comput. 43:256â279, 2014), DNF sparsification Gopalan et al. (Comput. Complex. 22:275â310 2013), randomness extractors Li et al. (In: APPROX-RANDOM, LIPIcs 116:51:1â13, 2018), and recent advances on the ErdĆs-Rado sunflower conjecture Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) Lovett et al. (From dnf compression to sunflower theorems via regularity, 2019) Rao (Discrete Anal. 8,2020). The recent breakthrough of Alweiss, Lovett, Wu and Zhang Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) gives an improved bound on the maximum size of a w-set system that excludes a robust sunflower. In this paper, we use this result to obtain an [Formula: see text] lower bound on the monotone circuit size of an explicit n-variate monotone function, improving the previous best known [Formula: see text] due to Andreev (Algebra and Logic, 26:1â18, 1987) and Harnik and Raz (In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, New York, 2000). We also show an [Formula: see text] lower bound on the monotone arithmetic circuit size of a related polynomial via a very simple proof. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an [Formula: see text] lower bound on the monotone circuit size of the CLIQUE function for all [Formula: see text] , strengthening the bound of Alon and Boppana (Combinatorica, 7:1â22, 1987)
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Discrete Fourier Analysis and Its Applications
The topic of discrete Fourier analysis has been extensively studied in recent decades. It plays an important role in theoretical computer science and discrete mathematics. One hand it is interesting to study the structure of boolean functions via discrete Fourier analysis. On the other hand, these structural results also provide a huge number of applications in theoretical computer science, including computational complexity, pseudorandomness, cryptography, learning theory. In this dissertation, we extend some more connections between discrete Fourier analysis and theoretical computer science. In particular, we study the following questions.\begin{itemize}\item Robust sensitivity of boolean function. In this part, we study the connection between the Fourier tail bound and the sensitivity tail bound of boolean functions, which is an analogue of the sensitivity conjecture, which was proposed by Nisan \cite{nisan1991crew}.\item DNF sparsification. The disjunctive normal form (or DNF) is a widely used representation of boolean functions. It is very interesting to study the structure of DNFs. There are two natural ways to measure the complexity of DNFs, the width and the size. In this thesis, we study a connection between these two measures. We propose a new approach by combing the swithing lemma (a combinatoric tool) and the hypercontrativity inequality (an analytic inequality). This framework does also suggest a new approach to the famous sunflower conjecture.\item Applications in learning theory. In 1989, the first Fourier-based learning algorithms was introduced by a seminar paper of Linial, Mansour and Nisan \cite{linial1989constant}. Followed by a series of subsequent works, people found that discrete Fourier analysis is powerful to design learning algorithms. One hand sparse Fourier functions are strong enough to approximate a lot of functions, on the other hand sparse Fourier functions are relatively easy to learn. Build on this framework, we give a more efficient algorithm to solve the \emph{population recovery} problem. That is how to recover a unknown distribution from noisy samples.\end{itemize
Kernelization of generic problems : upper and lower bounds
This thesis addresses the kernelization properties of generic problems, defined via syntactical restrictions or by a problem framework. Polynomial kernelization is a formalization of data reduction, aimed at combinatorially hard problems, which allows a rigorous study of this important and fundamental concept. The thesis is organized into two main parts. In the first part we prove that all problems from two syntactically defined classes of constant-factor approximable problems admit polynomial kernelizations. The problems must be expressible via optimization over first-order formulas with restricted quantification; when relaxing these restrictions we find problems that do not admit polynomial kernelizations. Next, we consider edge modification problems, and we show that they do not generally admit polynomial kernelizations. In the second part we consider three types of Boolean constraint satisfaction problems.We completely characterize whether these problems admit polynomial kernelizations, i.e.,given such a problem our results either provide a polynomial kernelization, or they show that the problem does not admit a polynomial kernelization. These dichotomies are characterized by properties of the permitted constraints.Diese Dissertation beschĂ€ftigt sich mit der Kernelisierbarkeit von generischen Problemen, definiert durch syntaktische BeschrĂ€nkungen oder als Problemsystem. Polynomielle Kernelisierung ist eine Formalisierung des Konzepts der Datenreduktion fĂŒr kombinatorisch schwierige Probleme. Sie erlaubt eine grĂŒdliche Untersuchung dieses wichtigen und fundamentalen Begriffs. Die Dissertation gliedert sich in zwei Hauptteile. Im ersten Teil beweisen wir, dass alle Probleme aus zwei syntaktischen Teilklassen der Menge aller konstantfaktor-approximierbaren Probleme polynomielle Kernelisierungen haben. Die Probleme mĂŒssen durch Optimierung ĂŒber Formeln in PrĂ€dikatenlogik erster Stufe mit beschrĂ€nkter Quantifizierung beschreibbar sein. Eine Relaxierung dieser BeschrĂ€nkungen gestattet bereits Probleme, die keine polynomielle Kernelisierung erlauben. Im Anschluss betrachten wir Kantenmodifizierungsprobleme und zeigen, dass diese im Allgemeinen keine polynomielle Kernelisierung haben. Im zweiten Teil betrachten wir drei Arten von booleschen Constraint-Satisfaction-Problemen. Wir charakterisieren vollstĂ€ndig welche dieser Probleme polynomielle Kernelisierungen erlauben. FĂŒr jedes gegebene Problem zeigen unsere Resultate entweder eine polynomielle Kernelisierung oder sie zeigen, dass das Problem keine polynomielle Kernelisierung hat. Die Dichotomien sind durch Eigenschaften der erlaubten Constraints charakterisiert
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum