Courses

In classic complexity theory, coping with NP-hard optimization problems often translates to the following fundamental question: “What is the best approximate solution that we can get in polynomial time?” We will present basic techniques for the design and analysis of approximation algorithms.
This includes combinatorial techniques (such as local search and local ratio) and techniques based on linear programming. We will demonstrate the use of these techniques for solving packing and scheduling problems, as well as problems on graphs.

Course covering a wide range of topics in complexity theory, such as space complexity, the polynomial hierarchy, probabilistic computation, interactive proofs, circuit complexity, and more.

We will discuss three main topics:

– Schaefer’s theorem, which shows that any Boolean CSP is either in P or NP-complete.
– Hardness of approximation: we will find the inapproximability thresholds for MAX-3LIN and for MAX-CUT.
– Proof complexity: we will show that some CNFs, including random ones, are hard to refute in Resolution.

If time permits, we will discuss a few more minor topics.

We will discuss the question of whether it is possible to design responsible algorithms that process sensitive data while avoiding the risk of exposing personal and sensitive information. We will address this question from a theoretical perspective, focusing on the concept of differential privacy, which offers a formal mathematical definition of algorithmic privacy. Course Content Overview:

• A systematic development of the definition of differential privacy and its basic properties.

• Design and analysis of differentially-private algorithms.

• Can any algorithmic task be solved privately? Analysis of the cost of differential-privacy in terms of computational and data efficiency.

• Additional selected topics as determined by the instructor’s discretion.

This is a theoretical course that demands mathematical maturity.

This course will introduce students to algorithmic foundations for handling large, high-dimensional datasets.

A sampling of topics and questions that will be covered includes:

• Streaming algorithms: How much data do we need to solve a problem whose input is revealed gradually?
• Dimensionality reduction: How can we decrease the dimension of the data while preserving its key properties?
• MapReduce algorithms: How can we correlate tens of thousands of machines to solve algorithmic problems?

Basic concepts: error correction, error detection and erasure correction. Linear error-correcting codes. Hamming codes. Introduction to finite fields. Reed-Solomon codes, BCH codes and alternant codes. Decoding algorithms. Bounds on the parameters of error-correcting codes.

Probabilistic methods have become a cornerstone of combinatorial theory, while probabilistic algorithms have become central to computer science.
This course introduces an array of already-classical and more modern probabilistic methods, along with their applications to math and CS. The course is centered on proof techniques and problem solving. Among the covered topics: basic methods, concentration inequalities, martingales, the local lemma, information entropy.

The seminar deals with distributed computing in modern computing systems, in the presence of different kinds of failures. Grades are based (mostly) on presenting in-depth a paper from the recent research literature.

The metric method is a powerful tool that has been used extensively in the last two decades in the design and analysis of algorithms.

This course will survey some of the basic techniques in the metric approach, as well as its applications to various topics such as: clustering and graph cuts, balanced graph partitioning, network routing, and online algorithms.

The topic of this course is uncertainty in algorithms. Given a computational problem, the traditional design and analysis of algorithms assumes that complete knowledge of the input is known in advance. However, this is not the case in many applications, including resource management and allocation, data structures, and game dynamics. In these scenarios making decisions in the face of uncertainty is of major importance. Take, for example, managing a cache memory; when there is a page miss, the decision which page to evict from the cache is made without knowing future requests for pages.

In the course we will go over advanced techniques developed in recent years for dealing with such settings, where information about the future is unavailable or limited. We will mainly focus on competitive analysis of online algorithms and online learning, and study surprising connections between them. We will see how techniques from linear programming and convex optimization are useful for developing online algorithms.

Advanced topics in theoretical cryptography.

This course will introduce students to advanced algorithms for dynamic graphs. This active research area focuses on questions of the following flavor: “How quickly can we solve algorithmic problems on graphs that change slowly over time? In particular, how much faster than recomputing from scratch after every change in the graph?” This question is central to problems with input that changes over time (for example, in GPS applications, closure and opening of roads changes the road network), but is also key to speeding up algorithms with static inputs (in much the same way that basic data structures can be used to speed up other classic algorithms). The course’s objective is to expose students to techniques and results for basic problems in the field (such as matching, shortest paths, minimum spanning trees, etc). As part of the course, the students will also be exposed to central tools in algorithm design and theory of computer science more broadly that are relevant to the course topic, including linear programming and duality, randomized algorithms and the multiplicative weight update method. The course evaluation will be based on homework and a final project, which includes reading a research paper, summarizing the paper and simplifying it and/or improving on it.

Basic course on the theory of random graphs, focusing on the Erdős–Rényi model. Based on the nice Introduction to Random Graphs by Frieze and Karoński.

Grading is based on homework and a final project.

Proof-systems are at the very heart of theoretical computer science. For example the P vs NP question asks whether every statement that someone can prove to you, you could have actually solved by yourself.

In this course we will discuss advanced proof-systems such as:
* Interactive proofs, which are a natural generalization of NP proofs in which the verifier can interact with the prover.
* Zero-knowledge proofs, which allow one to prove the correctness of a statement without revealing anything else.
* Probabilistically Checkable Proofs (PCPs), which are proofs that can be verified by reading only a small number of bits.
* Doubly efficient proofs, which allow one to delegate expensive computations to an untrusted server.

Requirements: roughly 3 homework assignments and an exam.

Communication complexity studies the number of bits that should be exchanged in order to perform a computation whose input is distributed among two or more parties. This is studied under a spectrum of computational models: deterministic, non-deterministic, randomized etc.
Beyond the immediate implications that such questions have to understanding distributed computations, we will present various surprising applications to seemingly unrelated questions, mostly in Complexity Theory.
The course will use methods from Combinatorics, Probability Theory and Algebra.

An introductory course to the topic of Algorithmic Game Theory, the study of algorithms coupled with incentives. In light of the growing influence of algorithms on the economy and society, algorithm design must take into account the interaction with strategic players on top of traditional considerations like runtime complexity. As a prime example think of eBay auctions, which solicit bids from self-interested buyers.
The first half of the course will be dedicated to mechanism design, the science of designing such online auctions and markets. In the second half we will discuss the fundamental notion of equilibrium, including reaching it through centralized computation and decentralized learning.

Time: Mondays 10:30-12:30 (lecture) + Mondays 12:30-13:30 (recitation by Konstantin Zabarnyi)

Basic topics in theoretical cryptography.

Machine learning often involves and affects people – a.k.a. “strategic agents” – who have interests, preferences and rights. This seminar covers recent research in the interdisciplinary questions that arise from this combination. Main topics include: Economic mechanisms and learning; strategic classification; incentivizing exploration; fairness in automatic decision making; and more.

Prerequisites: Algorithms 1
Useful but not required: Probability Theory, Learning Theory, Algorithmic Game Theory

The main topics of the course are Gödel’s incompleteness theorems (including an axiomatic system for arithmetic, basics of recursion theory and Church’s thesis), set theory (axiomatic and combinatorial foundations of set theory as a basis for math), and modal logic (Kripke models, completeness theorems and connection to computer science).

For more information, please contact Professor Michael Kaminski.

Zero-knowledge proofs enable one to prove correctness of a computational statement without revealing any additional information. Zero-knowledge proofs have had a fundamental impact on cryptography and complexity theory and very recently are starting to be deployed also in practice.
This course will cover various advanced aspects of zero-knowledge proofs including:
1. Definitions, basic constructions and ZKP for NP.
2. Characterization of statistical zero-knowledge.
3. Non-interactive zero-knowledge.
4. A study of various efficiency aspects of zero-knowledge proofs.

The course will cover advanced algorithmic techniques developed for solving various (discrete) optimization problems. We will discuss linear programming, semidefinite programming, convex programming, and more. We will see applications of these techniques to a variety of applications, e.g., developing approximation algorithms for NP-hard problems, coloring (optimally) perfect graphs. Connections to learning theory will also be discussed.

Requirements: grades will be determined based on homework assignments.

We will study basic mathematical models in machine learning, including supervised learning, online learning, interactive learning, and distribution learning.
We will focus on combinatorial characterizations of the complexity of learning.

The course requires mathematical maturity and will involve discussions about open problems and active research directions.

For student who attended the class with Shay in the last semester:
This semester we will cover different topics (in particular we will begin with the online learning model).
Also, notice the course number is different from that of last semester, so one can get credit for both classes.

Definability deals with the expressive power of description languages which are used in program specification, databases and axiomatic mathematics. These languages include Regular Expressions, First Order Logic, Relational Algebra, Second Order Logic and various Temporal and Modal Logics.
Computability deals with computations and the resources needed to execute them in various models of computations. These include Finite Automata, Turing Machines, Register Machines, and the like.
The course deals with the interplay between definability in various descriptive languages and computability with various computing devices. It brings together theoretical aspects of Database Theory, Computability Theory, Complexity Theory and Logic.

The course can be given in English if required. The slides are in English.

The course will focus on topics related to the origins of life and in particular the origins of the genetic code.
As we all know when we see a code – someone wrote it; With one exception – the genetic code.
Relying on the notion of “autocatalytic sets” we shall explore this topic.

The use of mathematical programming is of paramount importance to the design and analysis of algorithms in general, and approximation algorithms in particular. While linear programming (LP) is the prototypical type of mathematical programming that is utilized, non-linear mathematical programming is also vastly used. This course will focus mainly on semi-definite programming (SDP) and survey its basic uses in the design and analysis of approximation algorithms as well as its applications to various topics such as: graph coloring, satisfiability, graph cuts, and clustering.

Boolean function analysis is a set of tools coming from probability theory and functional analysis, with applications to theoretical computer science and combinatorics. The course is based, in large part, on the excellent monograph Analysis of Boolean functions by Ryan O’Donnell.

Grading is based on homework and a final project.

We will study basic mathematical models in machine learning, including supervised learning, online learning, interactive learning, and distribution learning.
We will focus on combinatorial characterizations of the complexity of learning.

The course requires mathematical maturity and will involve discussions about open problems and active research directions.

Prerequisites: Combinatorics and Probability Theory.

This is a seminar course on combinatorial optimization. The course will focus on two different methodologies: local ratio and primal-dual.
The first three lectures will serve as an introduction to approximation algorithms, via linear programming.
In the fourth lecture we will explore primal-dual techniques, and in the fifth the competing methodology, local ratio.
During the rest of the semester we will encounter many different applications to various optimization problems, such as vertex cover, set cover, Steiner trees, cycle cover, assignment problem, resource allocation, and more.

Property testing algorithms are used to decide if some mathematical object (such as a graph or a Boolean function) has a “global” property, or is “far” from having this property, using only a small number of “local” queries to the object.

This is a seminar course on combinatorial optimization. The course will focus on two different methodologies: local ratio and primal-dual.

Topics in pseudorandomness.

Discovery-based class, on a variety of topics. Students work in groups on handouts, with online help from the lecturer.

Grading based on final project.

Algorithms involving numbers, polynomials and matrices: FFT, fast integer multiplication, fast matrix multiplication, primality testing, integer factorization.