# Semináø z výpoèetní slo¾itosti

## Pavel Pudlák, Jiøí Sgall

### Pøedchozí program semináøe [Previous program]

26. 5. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Ronald de Wolf: Error-Correcting Data Structures. STACS'09.
(referuje Ondøej Zajíèek)

Abstract: We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been corrupted by a constant fraction of errors. This new model is the common generalization of (static) data structures and locally decodable error-correcting codes. The main issue is the tradeoff between the space used by the data structure and the time (number of probes) needed to answer a query about the encoded object. We prove a number of upper and lower bounds on various natural error-correcting data structure problems. In particular, we show that the optimal length of error-correcting data structures for the Membership problem (where we want to store subsets of size s from a universe of size n) is closely related to the optimal length of locally decodable codes for s-bit strings.

12. 5. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

E. Mossel, R. O'Donnell, and R. Servedio: Learning functions of k relevant variables.
(referuje Michal Koucky)

Abstract: We consider a fundamental problem in computational learning theory: learning an arbitrary Boolean function which depends on an unknown set of $k$ out of $n$ Boolean variables. We give an algorithm for learning such functions from uniform random examples which runs in time roughly $(n^k)^{{\frac \omega {\omega + 1}}},$ where $\omega < 2.376$ is the matrix multiplication exponent. We thus obtain the first polynomial factor improvement on the naive $n^k$ time bound which can be achieved via exhaustive search. Our algorithm and analysis exploit new structural properties of Boolean functions.

5. 5. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Benjamin Rossman: On the Constant-Depth Complexity of k-Clique
(referuje Phuong Nguyen)

Abstract: I will present a recent paper by Rossman showing a lower bound w(n^k/4) for constant-depth circuits that compute k-CLIQUE, for every fixed constant k. This is only a polynomial lower bound, but it doesn't depend on the depth of the circuits. The result implies that the variable fragments of first order logic (FO) on ordered graphs form a strict hierarchy, thus resolving a long standing problem in Finite Model Theory.

21. 4. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

H. Buhrman, P. B. Miltersen, J. Radhakrishnan, and S. Venkatesh: Are Bitvectors Optimal? SIAM J.Comput. 31:1663-1958, 2002.
(referuje Jiøí Sgall)

Abstract: We study the it static membership problem: Given a set S of at most n keys drawn from a universe U of size m, store it so that queries of the form "Is u in S?" can be answered by making few accesses to the memory. We study schemes for this problem that use space close to the information theoretic lower bound of $\Omega(n\log(\frac{m}{n}))$ bits and yet answer queries by reading a small number of bits of the memory.

We show that, for $\epsilon > 0$, there is a scheme that stores $O(\frac{n}{\epsilon^2}\log m)$ bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most $\epsilon$. We consider schemes that make no error for queries in S but are allowed to err with probability at most $\epsilon$ for queries not in S. We show that there exist such schemes that store $O((\frac{n}{\epsilon})^2 \log m)$ bits and answer queries using just one bitprobe. If multiple probes are allowed, then the number of bits stored can be reduced to $O(n^{1+\delta}\log m)$ for any $\delta > 0$. The schemes mentioned above are based on probabilistic constructions of set systems with small intersections.

We show lower bounds that come close to our upper bounds (for a large range of n and $\epsilon$): Schemes that answer queries with just one bitprobe and error probability $\epsilon$ must use $\Omega(\frac{n}{\epsilon\log(1/\epsilon)} \log m)$ bits of storage; if the error is restricted to queries not in S, then the scheme must use $\Omega(\frac{n^2}{\epsilon^2 \log (n/\epsilon)}\log m)$ bits of storage.

We also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes.

7. 4., 14. 4. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Ran Raz: Elusive Functions and Lower Bounds for Arithmetic Circuits. STOC 2008.
(referuje Pavel Pudlák)

Abstract: A basic fact in linear algebra is that the image of the curve f(x)=(x1,x2,x3,...,xm), say over C, is not contained in any m-1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree~1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f:C → Cm, such that, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness) implies super-polynomial lower bounds for computing the permanent over C. More generally, we say that a polynomial-mapping f:Fn → Fm is (s,r)-elusive, if for every polynomial-mapping Γ:Fs → Fm of degree r, Im(f) ⊄ Im(Γ). We show that for many settings of the parameters n,m,s,r, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every r < log n, we give an explicit example for a polynomial-mapping f:Fn → Fn2, of degree O(r), that is (s,r)-elusive for s = n1+Ω(1/r). We use this to construct for any r, an explicit example for an n-variate polynomial of total-degree O(r), with coefficients in {0,1,}such that, any depth r arithmetic circuit for this polynomial (over any field) is of size ≥ n1+Ω(1/r). In particular, for any constant r, this gives a constant degree polynomial, such that, any depth r arithmetic circuit for this polynomial is of size ≥ n1+Ω(1). Previously, only lower bounds of the type Ω(n • λr (n)), where λr (n) are extremely slowly growing functions (e.g., λ5(n) = log n, and λ7(n) = log* log*n), were known for constant-depth arithmetic circuits for polynomials of constant degree.

31. 3. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Stasys Junka: Representing (0,1)-matrices by depth-2 circuits with arbitrary gates
(referuje Michal Koucký)

Abstract: A boolean circuit represents an n by n (0,1)-matrix A if it correctly computes the linear transformation y = Ax over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than O(n^2/ln n) wires. We first show that using non-linear gates one can save a lot of wires: Any matrix can be represented by a depth-2 circuit with O(n ln n) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n (0,1)-matrix differ in at least d rows, then the matrix requires Omega(d ln n/ln ln n) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates.

24. 3. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Pavel Pudlák: On convex complexity measures III
(joint work with P. Hrube¹, S. Jukna, and A. Kulikov)

Abstract: I talked about this subject already twice. Now we are preparing the final version so it will be helpful to talk about it again. I will repeat some stuff, but we view the results from a different perspective now and there are some new results not covered in previous talks. I will also mention some observations and problems that will not be in the paper.

3. 3. 2009 (úterý [Tuesday], 14:30, MÚ ®itná 25, posluchárna v pøízemí zadní budovy)

Ketan Mulmuley: On P vs NP, Geometric Complexity Theory, and the Riemann Hypothesis
(video from Institute for Advanced Study, February 9, 2009, http://video.ias.edu/csdm/pvsnp)

Abstract: This series of three talks will give a nontechnical, high level overview of geometric complexity theory (GCT), which is an approach to the P vs. NP problem via algebraic geometry, representation theory, and the theory of a new class of quantum groups, called nonstandard quantum groups, that arise in this approach. In particular, GCT suggests that the P vs. NP problem in characteristic zero is intimately linked to the Riemann Hypothesis over finite fields. No background in algebraic geometry, representation theory or quantum groups would be assumed.