Bahman continued his presentation of the Survivable Network Design approximation algorithm. We got stuck on some issues about vertex solutions to the LP... :-\ |

A straggler talk in the Algorithms & Theory seminar: Janno talked about proximal and augmented Lagrangian algorithms for machine learning problems with loss and regularization terms in the objective function. |

Bahman started presenting, in all the dirty details, the iterated LP-rounding approximation algorithm for survivable network design. |

After the exam stress has subsided, we started our usual schedule again: Over the summer, Tuesdays will be dedicated to the theory behind Machine Learning. |

We worked through the proof that every hypergraph with sufficiently many edges has a 2-coloring with discrepancy at most sqrt( n log(m/n) ). Highlights of the proof included the use of entropy and Kleitman's theorem. |

Abdullah explained how the Lagrange multipliers form the sub-gradient or gradient, respectively, of a parameterized continuous optimization problem. |

Bahman continued his discussion of neural networks: a generator-discriminator pair sample from a given distribution. |

Dominique presented a problem related to quantum query complexity of random function inversions. |

Visiting PhD student Mahdi gave a survey over his research in edge coverings, vertex coverings: complexity, algorithms, combinatorics. He also discussed some of the problems he hopes to attach while in Tartu. |

Abdullah explained Lagrange duality for optimization problems: The Lagrange dual function, the Lagrange dual, weak duality theorem, strong duality, ... |

Bahman explained how to train feed-forward neural networks by gradient descent, using backpropagation. |

Theoretical Computer Science at the University of Tartu consists of the groups Crypto+Security, Coding Theory, and Algorithms & Theory, each with it's own Monday 2pm meeting, discussing their own specialized research. first Monday of the month, all of TCS meets for a "Joint Seminar". The idea is to exchange problems and ideas. People present ongoing research in an informal way--in a language that everybody in TCS understands, regardless of their specialization.Today, March 2, 2015, for the first ever Joint TCS Seminar, Ivo Kubjas talked about Index Coding, Ago-Erik Riet presented a question regarding growth of balls in permutation metrics, and Dominique Unruh started presenting issues regarding quantum non-interactive zero-knowledge proofs. The 2nd Joint TCS Seminar will take place on April 6 (room 215, 14:15). |

A sorting network can be thought of as solving the task of sorting a fixed number of inputs by a fixed network of wires and comparators. If realized as an electrical network, a sorting network can perform hardware level sorting. But sorting networks are also a new cool tool in Algorithms & Theory. On Monday, DOT will explain what sorting networks are, exactly, and discuss some applications in algorithms and combinatorics. The goal of this lunch session will be to understand the proof--using sorting networks--of a theorem by Kuhn et al on tournaments. |

Today, Janno Siim gave his seminar presentation on maximal m-free digraphs. He presented, for example, the proof for the fact every maximal m-free digraphs has an m-king. |

On Monday, DOT will present a theorem of Ronyai, Babai, and Ganapathy which gives an upper bound to the number of zero / nonzero patterns of a sequence of n polynomials in m variables of degree at most d. The perhaps most interesting fact is that the bound does not depend on n---only on m, d, and the number of non-zeros in the pattern. We will then discuss an application, due to Leslie Hogben and coauthors, to random matrix theory: the minimum rank of a matrix with a prescribed, random, zero-nonzero pattern. |

This week, Abdullah continued the proofs of the Karp's tail bounds on probabilistic recurrence relations. |

Today, Abdullah will talk about random recursive relations, and what can be said about expectation and upper tail based on very general assumptions. The main results go back to a paper of Karp from the 1990s. The motivation behind studying these relations lies in the analysis of randomized algorithms. |

This Monday, as on every first Monday of a month, we'll hold the joint TCS seminar. Probably Dominique will finish up his presentation of his quantum query complexity problem. After that, it's open stage. Post scriptum, added after the meeting. The following problems were discussed. DOT presented a probability/hypergraph problem connected to random k-SAT instances which he couldn't solve. Nobody had any useful ideas. Abdullah talked about an algorithmic problems on strings with applications in bioinformatics, which spawned a lively discussion. |

In today's Discrete Lunch, we'll discuss randomness extractors. A randomness extractor is a function which maps the points of a probability space to bit strings in such a way that, for all k, all length k strings are produced with equal probability (i.e., either 0 or 2.) One is then interested in the average length of the strings produced by the extractor.^{-k}We will first prove the easy upper bound on the average length of the strings produced by an extractor: the entropy of the probability space. After that, we'll look at a few constructions of randomness extractors. |

Tomorrow, April 6, 14:15, room 225, we'll hold the 2nd all-of-TCS seminar. Since the meeting is more "spontaneous" than the Discrete Lunch, I'll summarize it when it's over. -- Now here's the summary: Dominique continued presenting his problem about quantum interactive proofs. |

Today, for Discrete Lunch, we'll discuss a hypergraph coloring problem which arises from a question about thresholds for the satisfiability of random SAT instances. |

An assignment of one of two colors, red or blue, to every vertex of a hypergraph is called a proper 2-coloring, if no edge is monochromatic. Hypergraphs which have a proper 2-coloring are sometimes said to have Property B (as in "bipartite", maybe?).Unlike for usual graphs, where deciding whether it is bipartite can easily done in polynomial time, deciding whether a given hypergraph has a proper 2-coloring is an NP-complete problem. In today's Discrete Lunch, we'll discuss a smart re-coloring procedure, which finds a proper 2-coloring for every r-uniform hypergraph which does not have too many edges. |

Mozhgan will present the contents of that paper. Here's the abstract: We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix, and thus attract a special attention. |

In Monday's (March 23) Discrete Lunch, Abdullah will give a presentation about the probabilistic analysis of sorting algorithms. If one is interested in the situation where the input is a random permutation, this is a textbook topic (covered, e.g., in MTAT.05.008 Discrete Math, and MTAT.05.117 Randomness). Abdullah will present the contents of a paper by Clement, Nguyen, and Vallee, which considers the situation where the input is a random list of strings, which have to be sorted lexicographically, and one is interested in the number of comparisons of symbols. This setting is considerably more complex and requires more sophisticated probabilistic machinery. Abdullah will explain all of that in detail. |

On Monday, March 16, we'll proceed with discussing the Fourier transform on Z_n. Grabbing, en passant, Kronecker's theorem in diophantine approximation, we'll see one of the uniformity vs. density increment proofs for the existence of arithmetic progressions of length 3 in subsets of Z_n. |

For the Discrete Lunch on Monday March 9, 2015, DOT will review Fourier analysis over Z_n, and use it to show proofs of some theorems in additive combinatorics. |