Seminars

31.05.2017 12:15
Grzegorz Bukowiec
Computer science foundations
The Undecidability of the Generalized Collatz Problem by Stuart A. Kurtz and Janos Simon
The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. In this paper, we build on work of J. H. Conway to show that a natural generalization of the Collatz problem is $PI^0_2$ complete.
31.05.2017 16:15
Piotr Micek
Theoretical computer science
Ramsey Theory for Binary Trees and the Separation of Tree-chromatic Number from Path-chromatic Number

We propose a Ramsey theory for binary trees and prove that for every r-coloring of "strong copies" of a small binary tree in a huge complete binary tree T, we can find a strong copy of a large complete binary tree in T with all small copies monochromatic. As an application, we construct a family of graphs which have tree-chromatic number at most 2 while the path-chromatic number is bounded. This construction resolves a problem posed by Seymour.

Joint work with Fidel Barrera-Cruz, Stefan Felsner, Tamás Mészáros, Heather Smith, Libby Taylor, and Tom Trotter.

01.06.2017 16:15
Wojciech Kruk, Piotr Kruk
Combinatorial Optimization
Ulam Sequences and Ulam Sets
The Ulam sequence is given by a1=1,a2=2, and then, for n≥3, the element an is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence 1,2,3,4,6,8,11,13,16,…, which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors {v1,…,vk}⊂ℝn, we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.
07.06.2017 12:15
Jakub Nowak
Computer science foundations
Generic Complexity of Presburger Arithmetic by Alexander N. Rybalov
Fischer and Rabin proved in that the decision problem for Presburger Arithmetic has at least double exponential worst-case complexity (for deterministic and nondeterministic Turing machines). In paper 4  a theory of generic-case complexity was developed, where algorithmic problems are studied on "most" inputs instead of set of all inputs. An interesting question rises about existing of more efcient (say, polynomial) generic algorithm deciding Presburger Arithmetic on some set of closed formulas of
asymptotic density 1 (so-called generic set). We prove, however, that there is not even an exponential generic algorithm working correctly on a set of inputs which (so-called strongly generic set).
14.06.2017 16:15
Piotr Wójcik
Theoretical computer science
On the asymptotic density of valid sentences in first-order logic about one binary relation

This study arises from the following question: what is the proportion of tautologies of the given length n among the number of all FO relational sentences of length n? We investigate the simplest language with a fixed signature σ = {r}, where r is a binary relation symbol. The model with four logic symbols and an universal quantifier lead us to discover an unexpected result - the fraction of valid sentences is always greater than a fixed constant and therefore the density, if exists, is positive. The main theorem is derived from the analysis of structural properties of FO formulae, which themselves bear strict resemblance to structural properties of λ-terms.

Poprzednie referaty

25.05.2017 16:15
Sylwester Klocek, Maciej Woźniak
Combinatorial Optimization
On the complexity of the chip-firing reachability problem
In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. Both of these algorithms are strongly polynomial. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraph
24.05.2017 12:15
Piotr Wójcik
Computer science foundations
Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications by O.Bodini, J. David, and Ph. Marchal
In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of
random bit complexity, up to a multiplicative constant. We then apply a natural process “simulate-guess-and-proof” to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height
phenomenon.
23.05.2017 14:15
Szymon Policht
Cryptology
Supersingular isogeny key exchange
Supersingular isogeny is the newest addition to the post-quantum cryptography roster. It is elliptic curve based, but unlike tradidional ECC algorithms, it's quantum resistant. It offers significant key size reduction and computation time speedup compared to other post-quantum algorithms.
18.05.2017 16:15
Katrzyna Janocha
Combinatorial Optimization
Proper Orientations of Planar Bipartite Graphs
An orientation of a graph G is proper if any two adjacent vertices have different indegrees. The proper orientation number χ (G) of a graph G is the minimum of the maximum indegree, taken over all proper orientations of G. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that χ (G) ≤ 4, when G is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.
16.05.2017 14:15

Cryptology
Dzień Magistranta (sala 0004)
11.05.2017 16:15
Anna Kobak
Combinatorial Optimization
Lambda number for the direct product of some family of graphs
An L(2,1) labeling for a graph G = (V,E) is a function f on V such that | f(u) - f(v)| >= 2 if u,v are adjacent and f(u), f(v) are distinct if u,v are vertices of distance two. The lambda(G) for G is the minimum span over all L(2,1) labelings of G. We will show that when m>=6 and n>=3, lambda(Pm x Cn) = 7 if and only if n is not a multiple of 7 and also provide the conditions on m and n such that lambda(Cm x Cn) <= 7.
10.05.2017 16:15
Torsten Ueckerdt
Karlsruhe Institute of Technology
Theoretical computer science
The k-Strong Induced Arboricity of a Graph

Motivated by a connection to vertex-distinguishing colorings, we initiate the study of a new graph covering parameters: The k-strong induced arboricity. For a graph G and a positive integer k, a k-strong induced forest F in G is an induced forest in which every component has at least k edges. An edge in G is called k-valid if it is contained in at least one k-strong induced forest. The k-strong induced arboricity fk(G) is the smallest number m such that all k-valid edges of G can be covered with m k-strong induced forests in G.

We demonstrate that the k-strong induced arboricity is not monotone, neither in k, nor under taking subgraphs or even induced subgraphs. In particular we construct 2-degenerate graphs G with fk(G)
3 and fk+1(G) arbitrarily large. Focusing on the supremum fk(C) of fk(G) among all graphs G from a given graph class C, we prove that fk(G) is finite whenever C is of bounded expansion. For the class C of planar graphs we prove that fk(C) is 8,9 or 10. Moreover we give necessary and sufficient conditions for f1(C) to be finite in terms of r-shallow minors of graphs in C, a notion recently introduced by Nešetřil and Ossona de Mendez.

This is based on joint work with Maria Axenovich, Philip Dörr, Daniel Gonçalves, and Jonathan Rollin.

10.05.2017 12:15
Maciej Bendkowski
Computer science foundations
Analytic combinatorics: an introduction
In our talk we outline the main concepts and techniques of analytic combinatorics used to investigate properties of large random algebraic structures. We discuss the central interpretation of generating functions as functions analytic at the origin allowing to relate their analytic properties with the quantitative properties of studied structures. Finally, we briefly excerpt the techniques of singularity analysis allowing us to access the asymptotic form of corresponding counting sequences or investigate the probability distribution of interesting combinatorial parameters.
 
09.05.2017 14:15
Aleksandra Nowak
Cryptology
The Fully Homomorphic Encryption and Approximate Greatest Common Divisor Problem

We briefly introduce the definition of fully homomorphic encryption and describe the two main problems on which are based latest FHE schemes: The LWE/Ring-LWE and AGCD problems. We discuss their advantages and the relations between them. We present the definition of bootstrapping and investigate the FHE scheme based on the AGCD problem as published in [1].

References

[1] J. H. Cheon, D. Stehlé, Fully Homomorphic Encryption over the Integers Revisited, EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques, pp. 24--43.

04.05.2017 16:15
Grzegorz Bukowiec
Combinatorial Optimization
Even factors of graphs
An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. It has been shown that if a graph G has an even factor, it also has an even factor F such that |E(F)| >= 4/7 (|E(G)| + 1). 4/7 is the best possible ratio here, but we will try to strengthen this lower bound by taking the set of vertices of degree 2 into consideration.
27.04.2017 16:15
Jakub Szarawski
Combinatorial Optimization
A greedy approach to the Turtle Tower problem
In the Turtle Tower problem we are given n turtles with a mass and capacity for each of them. We are looking for the highest tower possible, regarding that capacity of every turtle in the tower cannot be exeeded by the sum of the masses of turles it carry. Presented solution is faster than commonly known dynamic one.
26.04.2017 16:15
Marcin Pilipczuk
University of Warsaw
Theoretical computer science
Subexponential Parameterized Algorithms for Planar Graphs, Apex-Minor-Free Graphs and Graphs of Polynomial Growth via Low Treewidth Pattern Covering

We prove the following theorem. Given a planar graph G and an integer k, it is possible in polynomial time to randomly sample a subset A of vertices of G with the following properties:

1) A induces a subgraph of G of treewidth O(√k log k), and

2) for every connected subgraph H of G on at most k vertices, the probability that A covers the whole vertex set of H is at least (2O(√k log2k)nO(1))−1, where n is the number of vertices of G.

Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound 2O(√k log2k)nO(1). The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph; examples of such problems include Directed k-Path, Weighted k-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface. We also provide a similar statement for graph classes of polynomial growth.

In the talk I will first focus on the background and motivation, and then highlight the main ideas of the proof by sketching the proof for the case of graph classes of polynomial growth. Based on joint work with Fedor Fomin, Daniel Lokshtanov, Dániel Marx, Michał Pilipczuk, and Saket Saurabh: http://arxiv.org/abs/1604.05999 and http://arxiv.org/abs/1610.07778.

26.04.2017 12:15
Konrad Kalita
Computer science foundations
Java Generics are Turing Complete by Radu Grigore
This paper describes a reduction from the halting problem of Turing machines to subtype checking in Java. It follows that subtype checking in Java is undecidable, which answers a question posed by
Kennedy and Pierce in 2007. It also follows that Java’s type checker can recognize any recursive language, which improves a result of Gil and Levy from 2016. The latter point is illustrated by a parser
generator for fluent interfaces.
25.04.2017 14:15
Michał Ziobro
Cryptology
Introduction to Homomorphic Encryption

The talk is divided into two parts. In the first part we briefly introduce Fully Homomorphic Encryption and a presentation of a classic example described in [1]. In the second part, we bring up a subject of partially homomorphic encrytpion schemes over finite fields, presented in [2].

References:

[1] C. Gentry, Computing Arbitrary Functions of Encrypted Data, 2008 (pdf)
[2] J. Liu, L. Chen and S. Mesnager, Partially homomorphic encryption schemes over finite fields, 2016 (pdf)

20.04.2017 16:15
Helena Borak, Zygmunt Łenyk
Combinatorial Optimization
Necklaces, Convolutions, and X + Y, A new upper bound for the online square packing
Necklaces, Convolutions, and X + Y

The necklace alignment problem is to find the optimal rotation of the necklaces to best align the beads, when we have two necklaces given, each with n beads at arbitrary positions. Alignment is measured according to the ℓ_p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = ∞ and how these problems can be reduced to convolution problems which can be solve in subquadratic time. Besides, we say how the necklace alignment problems, and their corresponding convolution problems, are also intrinsically connected to problems on X + Y matrices.

A new upper bound for the online square packing

In online square packing problem we try to minimise the height of squares on a plane with width 1. Squares come one by one, they can’t overlap and once set, it’s position can’t be changed. A new upper bound (ratio between algorithm result and optimal packing) is found by applying modified version of previously used First Fit Shelf algorithm.
19.04.2017 16:15
Lech Duraj, Adam Polak
Theoretical computer science
Longest Common Strictly Increasing Subsequecnce

The Longest Common Increasing Subsequence problem is a variant of classic Longest Common Subsequence problem, which can be solved in quadratic time with a simple dynamic programming algorithm. During the talk we will show a reduction from the Orthogonal Vectors problem to the Longest Common Increasing Subsequence problem which proves that the latter cannot be solved in strongly subquadratic time unless the SETH is false.

 

Simple modifications of the reduction prove that the problem for k sequences cannot be solved in O(nk-ε) time, that the same lower bounds apply to the Longest Common Weakly Increasing Subsequence, and that the assumption of SETH can be replaced with a weaker statement about satisfiability of non-deterministic branching programs.

12.04.2017 12:15
Jarek Duda
Computer science foundations
Boundaries for hashing problem, or how many bits ones individuality costs
I will talk about information-theoretic boundaries for the hashing problem, the Bloom filter, and generally about informational content of structures like trees and graphs. While the label size has to grow like logarithm of the population size, neglecting information about the order (lg(n!) bits), we get a linear growth of entropy of population, allowing to bound 'the cost of individuality' asymptotically to ~2.33275 bits per object.
06.04.2017 16:15
Andrzej Głuszyński, Jakub Nowak
Combinatorial Optimization
Local Antimagic Vertex Coloring of a Graph, A short proof of Cayley's tree formula
Local Antimagic Vertex Coloring of a Graph

The edge labelling is called 'local antimagic', if for all vertices sum of labels for incident edges is different for every two adjacent vertices. Such sum induce a correct vertex colouring. The local antimagic chromatic number - X_la(G) - is the minimum number of colours used by any proper local antimagic labelling. In the paper authors present results on this parameter for trees, friendship, wheel and clique graphs.


A short proof of Cayley's tree formula

Cayley’s tree formula is a very elegant result in Graph Theory. The problem is to find the number of all possible trees on a given set of labeled vertices. For n = 2 and vertex set {v1, v2}, we have only one tree. For n = 3 and vertex set {v1, v2, v3}, we have 3 different trees. Similarly for n = 4, we have 16 trees. We give a short proof of Cayley’s tree formula for counting the number of different labeled trees on n vertices.

Alok Bhushan Shukla, A short proof of Cayley's tree formula.
05.04.2017 12:15
Szymon Stankiewicz
Computer science foundations
CANTOR POLYNOMIALS AND THE FUETER-POLYA THEOREM by MELVYN NATHANSON

A packing polynomial is a polynomial that maps the set N^2 of lattice points with nonnegative coordinates bijectively onto N. Cantor constructed two quadratic packing polynomials, and Fueter and Polya proved analytically that the Cantor polynomials are the only quadratic packing polynomials.
The purpose of this paper is to present a beautiful elementary proof of Vsemirnov of the Fueter-Polya theorem. It is a century-old conjecture that the Cantor polynomials are the only packing polynomials on N^2.

04.04.2017 14:15
Mateusz Jachna
Cryptology
Secure Hash Algorithms family and the recently found collision for SHA-1
28.03.2017 14:15
Piotr Wójcik
Cryptology
Quantum Authentication with Key Recycling

We show that a family of quantum authentication protocols introduced in FOCS 2002 can be used to construct a secure quantum channel and additionally recycle all of the secret key if the message is successfully authenticated, and recycle part of the key if tampering is detected. We give a full security proof that constructs the secure channel given only insecure noisy channels and a shared secret key. We also prove that the number of recycled key bits is optimal for this family of protocols, i.e., there exists an adversarial strategy to obtain all non-recycled bits. Previous works recycled less key and only gave partial security proofs, since they did not consider all possible distinguishers (environments) that may be used to distinguish the real setting from the ideal secure quantum channel and secret key resource.

References:

[1] Christopher Portmann, Quantum Authentication with Key Recycling (pdf)

23.03.2017 16:15
Aleksandra Mędrek, Marcin Muszalski
Combinatorial Optimization
Planning for Fast Connectivity Updates
Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d > 1 deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linear-space representation of graphs which enables us to determine how a batch of edge updates can impact the graph. Given a set of d edge updates, in time O(d polylg n) we can obtain the number of connected components, the size of each component, and a fast oracle for answering connectivity queries in the updated graph. The initial representation is polynomial-time constructible.
  1. Mihai Patrascu , Mikkel Thorup, Planning for Fast Connectivity Updates, Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, p.263-271, October 21-23, 2007
21.03.2017 14:15
Jan Szczepaniec
Cryptology
Inclusive Block Chain Protocols

Distributed cryptographic protocols such as Bitcoin and Ethereum use the block chain to synchronize a global log of events between nodes in their network. Previous research has shown that the mechanics of the block chain and block propagation are constrained: if blocks are created at a high rate compared to their propagation time in the network, many conflicting blocks are created and performance suffers greatly.
Inclusive Block Chain Protocol is a alternative structure consists of a directed acyclic graph of blocks to the chain that allows for operation at much higher rates. It is showed that with this system the advantage of highly connected miners is greatly reduced. On the negative side, attackers that attempt to maliciously reverse transactions can try to use the forgiving nature of the DAG structure to lower the costs of their attacks.


References:
[1] Lewenberg Y., Sompolinsky Y., Zohar A., Inclusive Block Chain Protocols. In: Böhme R., Okamoto T. (eds) Financial Cryptography and Data Security, 2015. Lecture Notes in Computer Science, vol 8975. Springer, Berlin, Heidelberg (pdf)

16.03.2017 16:15
Patryk Urbański
Combinatorial Optimization
Generating Linear Extensions Fast

One of the most important sets associated with a poset P is its set of linear extensions, E(P). In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)), where e(P) = |E(P)|. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n*e(P)), where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y), for all pairs x,y, in time O(n^2 + e(P)).

Gara Pruesse, Frank Ruskey. Generating Linear Extensions Fast. SIAM J. Comput. Vol. 23, No. 2 (1994), pp. 373-386.

16.03.2017 14:15
Jakub Cisło, Grzegorz Jurdziński
Algorytmika
Tight Hardness Results for LCS and other Sequence Similarity Measures
15.03.2017 16:15
Manuel Bodirsky
TU Dresden
Theoretical computer science
The tractability conjecture for finitely bounded homogeneous structures
Finitely bounded homogeneous structures form a large class of infinite structures that can be seen as a generalisation of the class of all finite structures. Many results about finite structures, in particular in the context of the complexity of constraint satisfaction problems, can be generalised to this larger class. In this talk I will present a reformulation of a tractability conjecture for CSPs for this class in terms of polymorphisms, due to Barto and Pinsker (LICS 2016), and I will present a proof that the condition given in the tractability conjecture is decidable (under some Ramsey-theoretic assumptions that might hold in general for all reducts of finitely bounded homogeneous structures).
15.03.2017 12:15
Łukasz Lachowski
Computer science foundations
Impossibility of Distributed Consensus with One Faulty Process by MICHAEL J. FISCHER, NANCY A. LYNCH AND MICHAEL S. PATERSO
The consensus problem involves a asynchronous system of processes, some of which may be unreliable.The problem is for the reliable processes to agree on a binary value. In this paper, it is shown that every protocol for this problem has the possibility of nontermination, even with only one faulty process. By way of contrast, solutions are known for the synchronous case, the “Byzantine Generals” problem.
14.03.2017 14:15
Marcin Briański
Cryptology
Non-Interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers

The talk is based on the paper with the same title by Rosario Gennaro, Craig Gentry and Bryan Parno.

Verifiable Computation enables a computationally weak client to "outsource" the computation of a function F on various inputs x1, ..., xk to one or more workers. The workers return the result of the function evaluation, e.g., yi = F(xi), as well as a proof that the computation of F was carried out correctly on the given value xi. The verification of the proof should require substantially less computational effort than computing F(xi) from scratch.

We present a protocol that allows the worker to return a computationally sound, non-interactive proof that can be verified in O(m) time, where m is the bit-length of the output of F. The protocol requires a one-time pre-processing stage by the client which takes O(|C|) time, where C is the smallest Boolean circuit computing F. Our scheme also provides input and output privacy for the client, meaning that the workers do not learn any information about the values xi or yi.

09.03.2017 16:15
Grzegorz Matecki
Combinatorial Optimization
Boolean dimension of posets
A boolean dimension bdim(P) of a poset P=(X,<) is a smallest number k for which there is a set l1, l2, ..., lk of labelings X:->N and a boolean formula f(a1, ..., ak) such that the following is true: x < y in P iff f(a1, .., a_k) = 1 where ai =1 iff li(x) < li(x).
Generally, it is simple to observe that bdim(P) <= dim(P). Also, it is known that there is a constant c such that bdim(n) <= c log(n) for any poset P of size n.
The are few interesting open problems for boolean dimension:
1) Is boolean dimension of the boolean lattice of size n less that n?
2) Is there a constant c such that bdim(P) < c for any planar poset P?
 
09.03.2017 14:00
Sylwester Klocek, Wojciech Kruk
Algorytmika
The Alternating Stock Size Problem and the Gasoline Puzzle
08.03.2017 12:15
Maciej Bendkowski
Computer science foundations
Boltzmann samplers: random generation of combinatorial structures with an application to lambda calculus
In their seminal paper, Duchon et al. proposed a surprisingly simple, general-purpose framework of Boltzmann samplers meant for random generation of combinatorial structures. In this talk we revisit their method and discuss its elegant relation with analytic combinatorics as well as important practical implementation details. Finally, we discuss the application of Boltzmann samplers to the random generation of lambda terms used, e.g. in testing functional programming compilers.
07.03.2017 14:15
Zygmunt Łenyk
Cryptology
Speeding up modular multiplication using Montgomery and Barrett reduction
In the talk we present Montgomery and Barrett reductions that are used to speed up modular computations. In both reductions some pre-computations are made allowing for replacing subsequent expensive divisions by some fixed modulus with much cheaper operations involving a suitable power of 2. This is particularly useful when many modular divisions by the same modulus are performed (for example in finite field arithmetic or in RSA).
07.03.2017 14:15
Mateusz Twaróg, Łukasz Majcher
Combinatorial Optimization
Combinatorial library core
Presentation and discussion on core functionalities of the c++ combinatorial library. introduction to classes representing graphs, graph traversing algorithm templates and simple GUI.
01.03.2017 12:15
Michał Zwonek
Computer science foundations
Wielomianowe kodowania

Rozważany będzie problem istnienia wielomianowej bijekcji, najniższego stopnia, między N^k, a N. Przedstawione będą także problemy otwarte związane z tą tematyką.

Materiały do wystąpienia:

1) Elementarny dowód Twierdzenie Feuter-Polya (jedyny kwadratowy i bijektywny wielomian N^2->N to funkcja cantore'a) https://arxiv.org/abs/1512.08261

2) Praca, w której autorzy pokazują nieistnienie wielomianów 3 i 4 stopnia.  http://www.sciencedirect.com/science/article/pii/0022314X78900367

3) Praca podobnie tematyczna odnosząca się do problemu istnienia wielomianów bijektywnych z pewnego sektora N^2 w N. (To o czym wspomniałem na koniec, opis tego problemu jest też pod koniec w 1) ). Pod koniec pracy jest opisane 6 problemów otwartych związanych z tą tematyką. https://arxiv.org/abs/1305.2538

4) W podobnej tematyce. http://www.sciencedirect.com/science/article/pii/0022314X78900355
 

 

28.02.2017
Michał Dyrek
Cryptology
LLL algorithm and its applications in Number Theory and Cryptography
The talk is devoted to the algorithm by A. Lenstra, H. Lenstra and L. Lovász dated 1982 allowing for approximation of Shortest Vector Problem in polynomial time. We will present the idea of the algorithm and highlight its applications such as factoring polynomials over Q, constructing polynomials with small coefficients and connections with attacks on RSA.
26.01.2017 16:15
Wojciech Kruk, Maciej Woźniak
Combinatorial Optimization
A few open problems
We mentioned the following open problems in graph theory and discrepancy theory:

1. Erdos discrepancy problem
2. Hoang - Reed conjecture
3. Seagull problem - a consequence of Hadwiger's conjecture
25.01.2017 16:15
01.03.2017 16:15
Grzegorz Guśpiel
Theoretical computer science
Partial Visibility Representation Extension Problem

We study a class of graphs that have a special geometric representation. By a bar visibility representation of an undirected graph we mean a function that associates with each vertex of a graph a horizontal line segment in such a way, that between segments representing two ends of an edge there is a vertical strip (of visibility). In case of directed graphs, we additionally assume that the visibility is from the bottom to the top, that is the line segment representing the source of the edge is below the one for the target.

Graphs admitting such representations are well understood and can be recognized in linear time, both in the undirected and in the directed case. We work in a more subtle setting, where line segments are already associated with some vertices of a graph, and the question is if this can be extended to a bar visibility representation of an entire graph. We prove some results on complexity of this kind of problems.

This is joint work with Steven Chaplick, Grzegorz Gutowski, Tomasz Krawczyk and Giuseppe Liotta. The manuscript is available here: https://arxiv.org/abs/1512.00174

25.01.2017 12:00
Sylwester Klocek
Computer science foundations
Incompleteness, Undecidability and Automated Proofs by Cristian S. Calude and Declan Thompson
Incompleteness and undecidability have been used for many years as arguments against automatising the practice of mathematics. The advent of powerful computers and proof-assistants – programs that assist the development of formal proofs by human-machine collaboration – has revived the interest in formal proofs and diminished considerably the value of these arguments. In this paper we discuss some challenges proof-assistants face in handling undecidable problems – the very results cited above – using for illustrations the generic proof-assistant Isabelle.
24.01.2017
Kamil Sałaś
Cryptology
Lower Bounds for Discrete Logarithms
In the talk we will present the computational complexity of the discrete logarithm in the context of "generic algorithms", that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as unique binary string. For discrete logarithm, any generic algorithm must perform Ω(p^1/2) group operations, where p is the largest prime dividing the order of the group.
19.01.2017
Paweł Petecki
Akademia Górniczo-Hutnicza
Combinatorial Optimization
Symmetry breaking polynomial
Let G be a graph, and let Γ= Aut G. A coloring c of G is symmetry-breaking if for every non-identity automorphism φ in Γ, there is some vertex v of G such that c(v)≠c(φ(v)). There has been a lot of work on the minimum number of colors in a symmetry-breaking coloring of G. We discuss here a different problem: counting the number of k-colorings that are symmetry breaking. The tool, as is frequently the case for problems such as this one, is Möbius inversion. To solve this problem we define symmetry breaking polynomial ψG. For positive integer k (number of colors), ψG(k) is the number of k-colorings that break all non-trivial symmetries of the graph G.
18.01.2017 16:15
Marian Mrozek
Theoretical computer science
The discrete charm of Morse theory
The lecture will start with recalling P.S. Alexandroff's Theorem (1937) on mutual equivalence of posets and T0 topologies on finite sets. Next, we will outline the combinatorial version of the classical Morse Theory presented by R. Forman in 1998. Then, we will elaborate Forman's ideas towards the combinatorial topological dynamics with potential applications in Big Data problems and time series. The topics of the lecture will be expanded in a course for PhD students in the summer semester 2016/17.
18.01.2017 12:00
Michał Ziobro
Computer science foundations
Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search by Jose Espırito Santo, Ralph Matthes, Luıs Pinto
Kontynuacja seminarium z 23.11.2016
17.01.2017
Grzegorz Bukowiec
Cryptology
A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic

Until recently, all the algorithms for computing discrete logarithm had a sub-exponential complexity of type L(1/3), similar to the factorization problem. In this talk we'll discuss a heuristic algorithm that provides quasi-polynomial complexity for discrete logarithm in finite fields of small characteristic and that even for other cases still surpasses the Function Field Sieve method.

References:

[1] R. Barbulescu, P. Gaudry, A. Joux, E. Thomé, A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic (pdf)

11.01.2017 16:15
Patryk Mikos
Theoretical computer science
Online coloring of intervals with bandwidth
We study the online interval coloring problem with bandwidth. The input is a sequence of pairs Ji= (Ii,wi) where Ii is an interval on the real line and wi is a real number from (0,1]. In this setting a proper coloring is a function f:Ji →N such that for each color c and any point p on the real line, the sum of bandwidths of intervals containing p and colored by c does not exceed 1. The best known lower bound on the competitive ratio in this problem is 24/7. We present an explicit strategy for Presenter that increases the competitive ratio ifor this problem to at least 4.1626.
11.01.2017 12:00
Patryk Mikos
Computer science foundations
ON THE NUMBER OF DISTINCT LANGUAGES ACCEPTED BY FINITE AUTOMATA WITH n STATES by Michael Domaratzki, Derek Kisman and Jeffrey Shallit
We give asymptotic estimates and some explicit computations for both the number of distinct languages and the number of distinct finite languages over a k-letter alphabet that are accepted by deterministic finite automata (resp. nondeterministic finite automata) with n states.
10.01.2017
Szymon Policht
Cryptology
Faster operations on elliptic curves using Edwards curves
Elliptic curve cryptography is a broad and commonly used section of modern-day cryptography. Because of that, the speed of elliptic curve operations directly impacts the performance of many current systems. In this talk we'll show how to speed up those operations using Edwards curves.

References:
[1] Bernstein D.J., Lange T. (2007) Faster Addition and Doubling on Elliptic Curves. In: Kurosawa K. (eds) Advances in Cryptology – ASIACRYPT 2007. ASIACRYPT 2007. Lecture Notes in Computer Science, vol 4833. Springer, Berlin, Heidelberg (https://eprint.iacr.org/2007/286.pdf)
05.01.2017 14:15
Jan Derbisz, Jakub Łabaj
Algorytmika
How to sort by walking on a tree

We consider a tree with n vertices. On vertex number x there is a box with label p(x), with the function p being a permutation of {1,2,...,n}. A robot is walking on the tree, carrying at most one box at a time. If a box is placed where robot is standing, it can swap this box with the one being carried. The robot's goal is to sort the boxes, placing each one at the vertex with its number. The paper by D. Graf gives an algorithm computing the shortest possible robot's walk in quadratic time, as well as the proof that the problem becomes NP-complete if planar graphs are considered instead of trees.

04.01.2017 12:00
Konrad Kalita
Computer science foundations
ANALYTIC MODELS AND AMBIGUITY OF CONTEXT-FREE LANGUAGES by Philippe Flajolet
We establish that several classical context-free languages are inherently ambiguous by proving that their counting generating functions, when considered as analytic functions, exhibit some characteristic form of transcendental behaviour. To that purpose, we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages In particular, the paper contains a general density theorem for unambiguous context-free languages.
03.01.2017

Cryptology
(Cancelled)
22.12.2016 16:15
Łukasz Majcher, Jan Szczepaniec
Combinatorial Optimization
Convex p-partitions of bipartite graphs
A set of vertices X of a graph G is convex if no shortest path between two vertices in X contains a vertex outside X. We prove that for fixed p ≥ 1, all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time.
21.12.2016 16:15
Maciej Bendkowski
Theoretical computer science
Boltzmann samplers: a framework for random generation of combinatorial structures with an application to lambda calculus
In their seminal paper, Duchon et al. proposed a surprisingly simple, general-purpose framework of Boltzmann samplers meant for random generation of combinatorial structures. In this talk we revisit their method and discuss its elegant relation with analytic combinatorics as well as important practical implementation details. Finally, we discuss the application of Boltzmann samplers to the random generation of lambda terms used, e.g. in testing functional programming compilers.
20.12.2016 12:00
Bartłomiej Puget
Cryptology
An introduction to quantum computing and cryptography II
15.12.2016 14:15
Michał Glapa, Franciszek Stokowacki
Algorytmika
Maximum matching with algebraic methods

In 2006, a celebrated result by Mucha and Sankowski stated that the maximum matching problem can be done by Gaussian elimination. The complexity of this algorithm depends on matrix multiplication, but certainly beats O(n2.5) long-standing record of Micali-Vazirani algorithm.

15.12.2016
Anna Kobak
Combinatorial Optimization
Open problems in graph theory concerning minors.
We mentioned following open problems in graph theory:
  1. Hadwiger's Conjecture
  2. Seagull Conjecture
  3. Jorgensen's Conjecture
  4. Unfriendly partitions
  5. and a few more conjectures concerning minors.
14.12.2016 16:15
Grzegorz Matecki
Theoretical computer science
Two-Dimensional Irregular Packing Problem
We present results on packing irregular shapes onto given sheets of material.
14.12.2016 12:00
Piotr Wójcik
Computer science foundations
Enumeration and random generation of accessible automata by Frederique Bassino and Cyril Nicaud
We present a bijection between the A_n of deterministic and accessible automata with n states on a k-letters alphabet and some diagrams, which can themselves be represented as partitions of a set of kn + 1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of A_n is related to the Stirling number. Our bijective approach also yields an efficient random sampler, for the uniform distribution, of automata with n states, its complexity is O(n^3/2), using the framework of Boltzmann samplers.
13.12.2016 12:00
Krzysztof Kleiner
Cryptology
An introduction to quantum computing and cryptography I

In this talk we're going to discuss quantum informatics and its impact on the field of cryptography. We will introduce the basic concepts of quantum computing as well as cryptography based on Quantum Key Distribution scheme, one of the aspects of quantum informatics which already is being used in practice. Then we will present Shor's algorithm for polynomial-time factorization, responsible for the cryptosystems based on the hardness of factorization or discrete logarithm (in abelian groups) being no longer secure against an adversary with access to a quantum computer.


References:

[1] M. Hirvensalo, Quantum Computing (http://link.springer.com/book/10.1007/978-3-662-09636-9)
[2] F. Benatti, M. Fannes, R. Floreanini, D. Petritis, Quantum Information, Computation and Cryptography (http://link.springer.com/book/10.1007/978-3-642-11914-9)
[3] D. Deutsc, Lectures on Quantum Computation (http://www.daviddeutsch.org.uk/videos/)
[4] U. Vazirani, BerkeleyX's Lectures: Quantum Mechanics and Quantum Computation (https://www.edx.org/course/quantum-mechanics-quantum-computation-uc-berkeleyx-cs-191x)

08.12.2016
Lech Duraj
Algorytmika
A short tale of matrix multiplication

In recent years, some new algorithms for matrix multiplication problem were presented. Each of them is, however, only slightly faster than previous ones, while requiring substantially more complex analysis. Because of this, the long-standing question of optimal matrix multiplication algorithm seems even harder.

In my presentation, a short survery of the matrix multiplication algorithm is given. The presentation is based on François Le Gall's survey lecture of 2014.

08.12.2016
Zygmunt Łenyk
Combinatorial Optimization
Rendezvous on the line.
This is one of a handful of rendezvous problems where two players must find one another in a certain structured domain. In line case, players are placed on the line with distance 2, without knowing neither on which side is their partner, nor the direction of the line. I'll concentrate on the symmetric case where players must follow a specific (but maybe mixed) strategy. The conjecture is that best expected time of meeting two players equals 4.25.
  1.  
  2.  
  3. www.openproblemgarden.org/op/rendezvous_on_a_line
  4. Improved bounds for the symmetric rendezvous value on the line. Qiaoming Han, Donglei Du, Juan Vera, Luis F. Zuluaga. SODA 2007
07.12.2016 12:00
Jakub Brzeski
Computer science foundations
ENUMERATION OF FORMAL LANGUAGES by Michael Domaratzki
We survey recent results on the enumeration of formal languages. In particular, we consider enumeration of regular languages accepted by deterministic
and nondeterministic finite automata with n states, regular languages generated by regular expressions of a fixed length, and !-regular languages accepted by Müller automata. We also survey the uncomputability of enumeration of context-free languages and more general structures.
06.12.2016
Marek Rusinowski
Cryptology
Security of instant messaging applications.
Nowadays billions of people around the world are sharing sensitive information using instant messaging applications. We will look into the current state of IM security, the problems in this area and a few encryption protocols---OTR and Signal Protocol in particular---that provide security features desired by users.
01.12.2016
Aleksandra Mędrek, Krzysztof Maziarz
Algorytmika
Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back

The paper by Aleksander Mądry describes a new approach to the maximum flow problem. We define an electrical flow by assigning resistances to every edge and minimizing total energy instead of maximizing flow. Any flow network can be reduced to some electrical flow problem, using auxiliary reductions to some bipartite matching problems. The main result is a new maximum flow algorithm, working in O(m10/7). The presentation will cover a simpler, O(m3/2) version of the algorithm.

01.12.2016
Patryk Urbański
Combinatorial Optimization
Coloring Ordinary Maps, Maps of Empires and Maps of the Moon
A short review of generalized map coloring problems:
  • Heawood's empire coloring problem in the plane - 6M colors are sufficient to color a map of empires each consisting of at most M connected regions.
  • Earth-Moon map coloring Mathematics Magazine Vol. 66, No. 4 (Oct., 1993), pp. 211-226problem - it is known that the chromatic number of thickness-2 graphs is between 9 and 12. It is an open problem to find the exact value.
Coloring Ordinary Maps, Maps of Emipres, and Maps of the Moon. Joan P. Hutchinson. Mathematics Magazine. Vol. 66, No. 4 (Oct., 1993), pp. 211-226.
01.12.2016
Mateusz Twaróg
Combinatorial Optimization
Second Neighborhood via First Neighborhood in Digraphs

Second Neighborhood via First Neighborhood in Digraphs. Guantao Chen, Jian Shen, Raphael Yuster. Annals of Combinatorics. June 2003, Volume 7, Issue 1, pp 15–20.

30.11.2016 18:15
Bartosz Walczak
Theoretical computer science
Coloring curves that cross a fixed curve
A class of graphs is χ-bounded if the chromatic number of all graphs in the class is bounded by some function of their clique number. String graphs are intersection graphs of curves in the plane. Significant research in combinatorial geometry has been devoted to understanding the classes of string graphs that are χ-bounded. In particular, it is known since 2012 that the class of all string graphs is not χ-bounded. We prove that for every integer t≥1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is χ-bounded. This result is best possible in several aspects; for example, the upper bound t on the number of crossings with the fixed curve cannot be dropped. As a corollary, we obtain new upper bounds on the number of edges in so-called k-quasi-planar topological graphs. This is joint work with Alexandre Rok.
30.11.2016 12:00
Yauheni Ananchuk
Computer science foundations
ALGEBRAIC FOUNDATIONS FOR QUALITATIVE CALCULI AND NETWORKS by ROBIN HIRSCH, MARCEL JACKSON, AND TOMASZ KOWALSKI
Binary Constraint Problems have traditionally been considered as Network Satisfaction Problems over some relation algebra. A constraint network is satisfable if its nodes can be mapped into some representation of the relation algebra in such a way that the constraints are preserved. A qualitative representation is like an ordinary representation, but instead of requiring (a ; b) = a j b , as we do for ordinary representations, we only require that. A constraint network is qualitatively satisfable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfable then it is clearly qualitatively satisfable, but the converse can fail.
However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfable if and only if it is qualitatively satisfable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over
ordinary representations: whereas many finite relation algebras have only infnite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always ; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebra
29.11.2016
Anna Kobak
Cryptology
Breaking RSA vs Factoring in generic ring model

In the talk we present results of Aggarwal and Maurer [1], who showed that a generic ring algorithm for breaking RSA with modulus $N$ can be converted into an algorithm for factoring $N$. The results imply that any attempt at breaking RSA without factoring $N$ will be non-generic and hence will have to manipulate the particular bit-representation of the input modulo $N$. This provides new evidence that breaking RSA may be equivalent to factoring the modulus.

 

References:

[1] D. Aggarwal, U. Maurer, Breaking RSA Generically is Equivalent to Factoring, EUROCRYPT 2009

24.11.2016
Wojciech Łopata
Combinatorial Optimization
Several open problems from game theory, graph theory and combinatorics.

I'll briefly introduce the audience to two unrelated areas: book embedding and mechanism design, and walk through some open problems in those areas.
Wikipedia: Book embedding
Wikipedia: Mechanism design

24.11.2016
Dominika Salawa, Jakub Cisło
Algorytmika
Greedy algorithms for Steiner forest

The paper, resolves a long-standing open question: is the greedy approach for Steiner forest problem optimal up to multiplicative constant? The result by Anupam Gupta and Amit Kumar proves that in fact it is, with constant being at most 96. While some approximation algorithms for this problem, using linear programming, were previously known, this is the first known bound for a greedy algorithm.

23.11.2016
Piotr Danilewski
Theoretical computer science
Functional Code Building

 
A typical language translator uses a parser to convert an input string into some internal representation, usually in a form of an AST. The AST is then analyzed and transformed in passes. In the final pass, the AST is converted into the output, be it a machine code or source code of another language.
 
However, if we try to combine different languages, e.g. for a purpose of a multi-domain project, we may have a problem: each language uses different kinds of AST nodes, has different assumptions on the AST structure and features different pass algorithms. Settling for the common ground by limiting the node types, assumptions, and algorithms limits how the languages may reason about their code. Any high-level information is lost in such representation and high-level transformations cannot be defined.
 
Working on the common AST is similar to unstructured programming: AST acts as a global machine state. Any change to the state, e.g. introduced by a new language, may inadvertely affect the behavior of the existing languages. In regular programming we have moved away from unstructured programming towards higher abstraction levels, e.g. through functional programming. If it can be done to regular programming, can it be done to AST and code builders as well?
 
Our solution treats AST nodes not as objects of a fixed type. Instead, it is a function with its body describing entirely the behavior of such node. Even the connection between the nodes is represented internally by the function, in a form of continuations. 
The passes over the AST are replaced by a single invocation of these functions. The node functions may contain code for multiple stages of code analysis. In order to reduce the run time these additional stages are scheduled through dynamic staging.
 
This gives the power of abstraction to code building. Even nodes come from different languages may interact, as long as they understand the passed arguments.
 
This major change on how a code is represented requires changes in how we think about common basic operations during language interpretation:
- how do we link two nodes/functions together?
- how do we create control flow structures?
- how do we perform name lookups, even across different languages?
- how do we optimize the code so that the AST layer dissapears when generating code?
- what language-specific optimizations can be performed and how do we specify them?
 
We will address these questions in the upcoming talk.

23.11.2016
Michał Ziobro
Computer science foundations
Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search by Jos´e Espırito Santo, Ralph Matthes, Luıs Pinto
A new, comprehensive approach to inhabitation problems in simply-typed lambda-calculus is shown, dealing with both decision and counting problems. This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently. The representation may be seen as extending the Curry-Howard representation of proofs by lambda-terms, staying within the methods of lambda-calculus and type systems. Our methodology reveals inductive descriptions of the decision problems, driven by the syntax of the proof-search expressions, and the end products are simple, recursive decision procedures and counting functions.
22.11.2016
Aleksandra Nowak
Cryptology
Lattice Cryptography and The Learning With Errors Problem
17.11.2016
Paweł Kubiak
Combinatorial Optimization
Succinct Data Structures
17.11.2016
Patryk Gołębiowski, Wojciech Kruk
Algorytmika
Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms
In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. The paper by Colin White analyzes some of these algorithms and gives lower bounds for their complexity.
16.11.2016
Bartłomiej Bosek
Theoretical computer science
Every digraph is majority 4-choosable
A majority coloring of a digraph is a coloring of its vertices such that for each vertex at most half of its out-neighbors has the same color as that vertex. A digraph D is majority k-choosable if for any assignment of color lists of size k to the vertices there is a majority coloring of D from these lists. We prove the statement in the title. This gives a positive answer to a question posed recently in 1. This is a joint work with Marcin Anholcer and Jarosław Grytczuk.
  1. S. Kreutzer, S. Oum, P. Seymour, D. van der Zypen, D. R. Wood, Majority Colourings of Digraphs, Arxiv.
  2. Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk Every digraph is majority 4-choosable, Arixv.