slideshow 3

Complexity seminar

usually takes place each Friday at 13:30 in IM, rear building, ground floor
Chair: Michal Koucky, Pavel Pudlak
The programme is announced via the mailing list.

Testing Equality in Communication Graphs

Klim Efremenko
Tel-Aviv University
Friday, 19. May 2017 - 13:30 to 15:00

in IM, rear building, ground floor

Let $G=(V,E)$ be a connected undirected graph with $k$ vertices. Suppose that on each vertex of the graph there is a player having an $n$-bit string. Each player is allowed to communicate with its neighbors according to a (static) agreed communication protocol and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem? We determine this minimum up to a lower order additive term in many cases (but not for all graphs).   In particular, we show that it is $kn/2+o(n)$ for any Hamiltonian $k$-vertex graph, and that for any $2$-edge connected  graph with $m$ edges containing no two adjacent vertices of degree exceeding $2$ it is $mn/2+o(n)$. The proofs combine graph theoretic ideas with tools from additive number theory.

Cluster Planarity

Filip Šedivý
Faculty of Mathematics and Physics, Charles Univ.
Friday, 7. April 2017 - 13:30 to 15:00

in IM, rear building, ground floor

This thesis studies the problem of clustered planarity and follows two directions. First direction deals with computational complexity, where we show how clustered planarity can be solved in linear nondeterministic time in terms of the number of vertices of the input graph. Second directions is characterization of restricted versions of clustered planarity using minimal non-clustered-planar instances. For this purpose we introduce a notion of clustered minor using several operations reducing clustered graphs. We show that these operations preserve clustered planarity. We show that in the case of clustered graphs where clusters have size 2 and the graph is a cycle or a path, there are only finitely many minimal non-clustered-planar minors. We also mention known results about the computational complexity of clustered planarity.

Complexity theory beyond deterministic exponential time and applications Parts II

Igor Carboni Oliveira
Friday, 17. March 2017 - 13:30 to 15:00

in IM, rear building, ground floor

Part I. (this will be on the Seminar on limits of efficient computation on Marh 15) I'll survey a variety of connections between non-uniform circuit lower bounds, non-trivial proof systems, non-trivial learning algorithms, and variants of natural properties. In particular, I plan to discuss how one can prove new unconditional lower bounds for standard complexity classes such as REXP, ZPEXP, BPEXP, and NEXP by designing algorithms with a non-trivial running time, and to present some open problems.  Part II. I'll focus on a recent application of some of the complexity-theoretic techniques behind these results. I'll describe in more detail recent progress on the problem of efficiently generating large canonical prime numbers, via pseudodeterministic pseudorandomness. If there is enough time and interest, we will also discuss some non-constructive aspects of the result, and connections to derandomization. Based on joint work with Rahul Santhanam.

Lower bounds on Non-adaptive Data Structures for Median and Predecessor search

Anup Rao
Univ. of Washington, Seattle, USA
Friday, 10. March 2017 - 13:30 to 15:00

in IM, rear building, ground floor

What is the best way to maintain a subset of {1,2,...,m} so that the median of the set can be easily recovered? We are interested in the algorithms that access the fewest memory locations when adding an element to the set and computing the median of the set. We prove the first lower bounds for such algorithms. We say that the algorithm handles insertions non-adaptively if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. If the algorithm handles insertions non-adaptively, then our lower bounds imply that Binary Search Trees are essentially optimal (our results give a more general tradeoff between the parameters of the algorithm). In addition, we investigate the predecessor search problem, where the algorithm is required to compute the predecessor of any element in the set. Here we prove that if computing the predecessors can be done non-adaptively, then again Binary Search Trees are essentially optimal. Our... more