Seminars

03.06.2020 12:15
Ruslan Yevdokymov
Computer science foundations
Learnability can be undecidable by Shai Ben-David, Pavel Hrubes, Shay Moran, Amir Shpilka and Amir Yehudayoff
The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression.
04.06.2020 16:15
Raja L'hamri
Mohammed V University
Combinatorial Optimization
Examples of codes from zero-divisor graphs

In 2013, Dankelmann, Key, and Rodrigues introduced and investigated codes from incidence matrices of a graph. Several authors have been developed their study to several context. In this talk, we present some properties of codes associated with zero divisor graphs. Recall, the zero divisor graph of R denoted by Γ(R), is the simple graph associated with R whose set of vertices consists of all nonzero zero-divisors of R such that two distinct vertices x and y are joined by an edge if xy = 0. This is joint work with K. Abdelmoumen, D. Bennis, and F. Taraza.

  1. D. Bennis, J. Mikram, and F. Taraz. On the extended zero divisor graph of commutative rings. Turkish Journal of Mathematics. 40, 376-388, 2016.
  2. P. Dankelmann, J. D. Key, and B. G. Rodrigues. Codes from incidence matrices of graphs. Des. Codes Cryptogr. 68, 373-393, 2013.
  3. D. F. Anderson, P. S. Livingston. The Zero-Divisor Graph of a Commutative Ring. Journal of Algebra. 217(2), 434-447 , 1999.
  4. Raja L'hamri.Codes from zero-divisor graphs. slides. 2020.

(the seminar will only be online)

10.06.2020 12:15
Wojciech Grabis
Computer science foundations
Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup
This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|_gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.
10.06.2020 12:15
Wojciech Grabis

Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup
This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.
10.06.2020 12:15
Wojciech Grabis

Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup
This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|_gen to be the minimal size of all finite deterministic automata of genus g(L) computing L. We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|_gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.
10.06.2020 12:15
Wojciech Grabis

Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup
This article continues the study of the genus of regular languages that the authors introduced in a 2013
paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we
introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L)
computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily
far away from a finite deterministic automaton realizing the minimal genus and computing the
same language, in terms of both the difference of genera and the difference in size. In particular, we show
that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of
every regular language to be computable. This conjecture implies in particular that the planarity of a regular
language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the
conjecture for a fairly generic class of regular languages having no short cycles. The methods developed
for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we
show a new family of regular languages on a two-letter alphabet having arbitrary high genus.
01.01.2021 16:15
Grzegorz Gutowski
Theoretical computer science
TBA - Gutowski
01.01.2021 16:15
Bartosz Walczak
Theoretical computer science
TBA - Walczak
01.01.2021 16:15
Paweł Idziak
Theoretical computer science
Modular circuits satisfiability of intermediate complexity

In our paper [LICS'18] a generalization of Boolean circuits to arbitrary finite algebras was introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However nilpotent but not supernilpotent algebras have not been put on any side of this borderline.

This paper provides a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce Boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general.

Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy was shown by Bulatov and Zhuk.

Joint work with Piotr Kawałek and Jacek Krzaczkowski

01.01.2021 16:15
Krzysztof Turowski
Theoretical computer science
Degree Distribution of Dynamic Graphs Generated by a Duplication-Divergence Models
We analyze the structure of dynamic graphs generated from duplication models in which a new vertex selects an existing vertex and copies some of its neighbors and then we add some random divergence. We analyze various graph parameters like mean degree, number of open triangles, number of triangles, number of vertices of degree k or maximum degree in a graph generated from such models. We provide asymptotic analysis of expected values and tail behavior of these parameters. We also point to further extensions of this approach towards computing symmetries in these graphs and algorithms for graph compression.
 
Joint work with Philippe Jacquet, Alan Frieze and Wojciech Szpankowski

Poprzednie referaty

28.05.2020 17:00
Michał Stobierski
Combinatorial Optimization
The 1-2-3 Conjecture
  1. Michał Stobierski. The 1-2-3 Conjecture. slides. 2020.

(the seminar will only be online)

28.05.2020 16:15
Rafał Byczek
Combinatorial Optimization
Wegner’s conjecture - colouring the square of a planar graph

The square G2 of a graph G is the graph with the same vertex set in which two vertices are joined by an edge if their distance in G is at most two. The chromatic number of the square of a graph G is between D + 1 and D+ 1, where D is the maximum degree of G. Equivalently, the square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors.

In 1977, Gerd Wegner proved that the square of cubic planar graphs is 8-colorable. He conjectured that his bound can be improved - the chromatic number of G2 is at most 7, if D = 3, at most D + 5, if 4 ≤ D ≤ 7, and [3D / 2] + 1, otherwise. Wegner also gave some examples to illustrate that these upper bounds can be obtained.

C. Thomassen (2006) proved the conjecture is true for planar graphs with D = 3. The conjecture is still open for planar graphs with D ≥ 4. However several upper bounds in terms of maximum degree D have been proved as follows. In 1993, Jonas proved that χ(G2) ≤ 9D-19, for planar graphs with D ≥ 5. Agnarsson and Halldorson showed that for every planar graph G with maximum degree D ≥ 749, χ(G2) ≤ [9D / 5] + 2. Van den Heuvel and McGuinness (2003) showed that χ(G2) ≤ 2D + 25, Bordin (2002) proved that χ(G2) ≤ [9D / 5] + 1, if D ≥ 47, and Molloy and Salavatipour (2005) proved χ(G2) ≤ [5D / 3] + 78, moreover, χ(G2) ≤ [5D / 3] + 25 if D ≥ 241. Moreover, conjecture is confirmed in the case of outerplanar graphs and graphs without K4 minor.

The aim of the seminar will be to present the main facts about Wegner’s conjecture and main techniques and ideas which were used to prove some upper bounds. The presentation will be based on my master thesis.

  1. Rafał Byczek. Wegner’s conjecture Colouring the square of a planar graph. slides. 2020.

(the seminar will only be online)

27.05.2020 12:00
Szymaon Kapała
Computer science foundations
Searching for shortest and least programs by Cristian Caludea, Sanjay Jain, Wolfgang Merkle and Frank Stephan
The Kolmogorov complexity of a string x is defined as the length of a shortest program p of x for some appropriate universal machine U, that is, U(p) =x and p is a shortest string with this property. Neither the plain nor the prefix-free version of Kolmogorov complexity are recursive but for both versions it is well-known that there are recursive exact Solovay functions, that is, recursive upper bounds for Kolmogorov complexity that are infinitely often tight. Let a coding function for a machine M be a function f such that f(x) is always a program of x for M. From the existence of exact Solovay functions it follows easily that for every universal machine there is a recursive coding function that maps infinitely many strings to a shortest program. Extending a recent line of research, in what follows it is investigated in which situations there is a coding function for some universal machine that maps infinitely many strings to the length-lexicographically least program. The main results which hold in the plain as well as in the prefix-free setting are the following. For every universal machine there is a recursive coding function that maps infinitely many strings to their least programs. There is a partial recursive coding function (defined in the natural way) for some universal machine that for every set maps infinitely many prefixes of the set to their least programs. Exactly for every set that is Bennett shallow (not deep), there is a recursive coding function for some universal machine that maps all prefixes of the set to their least programs. Differences between the plain and the prefix-free frameworks are obtained by considering effective sequences I_1, I_2, ...of mutually disjoint finite sets and asking for a recursive coding function for some universal machine that maps at least one string in each set I_n to its least code. Such coding functions do not exist in the prefix-free setting but exist in the plain setting in case the sets I_n are not too small.
21.05.2020 16:15
Wojtek Grabis
Combinatorial Optimization
Algorithms for Destructive Shift Bribery.

Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods.

  1. Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. arXiv:1810.01763. 2018.
  2. Wojtek Grabis. Conflict-free Replicated Data Types. slides. 2020.

(the seminar will only be online)

20.05.2020 12:15
Piotr Mikołajczyk
Computer science foundations
Lambda Calculus and Probabilistic Computation by Claudia Faggian and Simona Ronchi della Rocca
We introduce two extensions of the lambda -calculus with a probabilistic choice operators, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic ! which allows us to develop a unified, modular approach.
14.05.2020 17:00
Jan Mełech
Combinatorial Optimization
Upper Bounds for the domination numbers of graphs

Sharareh Alipour and Amir Jafari showed various upper bounds for minimal cardinality of (a,b)-dominating set. For positive integers a and b, a subset S ⊆ V(G) is an (a,b)-dominating set if every vertex v ∈ S is adjacent to at least a vertices inside S and every vertex v ∈ V\S is adjacent to at least b vertices inside S. To achieve upper bounds, the authors used Turan's Theorem and Lovasz Local Lemma. These tools allowed them to obtain well-known bounds in a simpler way or new improved bounds in some special cases, including regular graphs.

  1. Sharareh Alipour, Amir Jafari. Upper Bounds for the Domination Numbers of Graphs Using Turán’s Theorem and Lovász Local Lemma. Graphs and Combinatorics35, 2019, 1153-1160.
  2. Jan Mełech. Upper bounds for domination number. slides. 2020.

(the seminar will only be online)

14.05.2020 16:15
Szymon Kapała
Combinatorial Optimization
Goldbach conjectures (weak and strong).

(the seminar will only be online)

13.05.2020 12:15
Przemysław Simajchel
Computer science foundations
Dance of the Starlings by Henk Barendregt, Jorg Endrullis, Jan Klop and Johannes Waldmann
In this birdwatching paper our binoculars are focused upon a particular bird from Smullyan's enchanted forest of combinatory birds, to wit the Starling. In the feathers of lambda -calculus this bird has the plumage \abc:ac(bc). This term is usually named S, reminiscent of its inventor Schonfinkel and also the combinatory ornithologist Smullyan. The combinator S is important for a variety of reasons. First, it is part of the \{ S, K\} basis for Combinatory Logic (CL). Second, there are several interesting questions and observations around S, mostly referring to termination and word problems. Our paper collects known facts, but poses in addition several new questions. For some of these we provide solutions, but several tough open questions remain.
07.05.2020 16:15
Michał Zwonek
Combinatorial Optimization
3-flow conjecture

3-flow-conjecture
“Every 4-edge-connected graph has a nowhere-zero 3-flow.”

Grötzsch proved that every triangle free (and loopless) planar graph is 3-colorable. By flow/coloring duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. The 3-flow conjecture asserts that this is still true without the assumption of planarity.

Jaeger proved that 4-edge-connected graphs have nowhere-zero 4-flows. The following weak version of the 3-flow conjecture used to remain open until 2010, but the original 3-flow conjecture remains wide open.

C̶o̶n̶j̶e̶c̶t̶u̶r̶e̶ (The weak 3-flow conjecture (Jaeger))
“There exists a fixed integer k so that every k-edge-connected graph has a nowhere-zero 3-flow.” 

These problems and the surrounding results will be presented during the seminar.

  1. Carsten Thomassen. The weak 3-flow conjecture and the weak circular flow conjecture. Journal of Combinatorial Theory, Series B102(2), 2012, 521-529.
  2. Fuyuan Chen, Bo Ning. A note on nowhere-zero 3-flow and Z_3-connectivity. arXiv:1406.1554. 2014.
  3. Michał Zwonek. Nowhere Zero Flow and related open problems, slides. 2020.

(the seminar will only be online)

06.05.2020 12:15
Bartłomiej Puget
Computer science foundations
Evidence Normalization in System FC by Dimitrios Vytiniotis and Simon Peyton Jones
System FC is an explicitly typed language that serves as the target language for Haskell source programs. System FC is based on System F with the addition of erasable but explicit type equality proof witnesses. Equality proof witnesses are generated from type inference performed on source Haskell programs. Such witnesses may be very large objects, which causes performance degradation in later stages of compilation, and makes it hard to debug the results of type inference and subsequent program transformations. In this paper, we present an equality proof simplification algorithm, implemented in GHC, which greatly reduces the size of the target System FC programs.
30.04.2020 17:00
Mateusz Kaczmarek
Combinatorial Optimization
χ-boundedness

If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? To answer that question Paul Seymour and Alex Scott took a closer look at recent progress in this field in their χ-boundedness survey. Based on their work I will present some results on forests and holes and few open problems like Gyárfás-Sumner conjecture or χ-boundedness connection to Erdős-Hajnal conjecture.

  1. Alex Scott, Paul Seymour. A survey of χ-boundedness. arXiv:1812.07500. 2018.
  2. Mateusz Kaczmarek. χ-boundedness, slides. 2020.

(the seminar will only be online)

30.04.2020 16:15
Kornel Dulęba
Combinatorial Optimization
Odd Perfect numbers

A number is perfect if it is equal to the sum of its divisors. So far only even perfect numbers have been found. For example, it was proven that squares of Mersenne’s numbers are perfect. However, no one has been able to prove that odd perfect numbers don’t exist. I’m going to start by presenting a summary of known facts about odd prime numbers. Then I’ll prove that an odd perfect number with at least eight distinct prime factors has to be divisible by 5.

  1. John Voight. On the nonexistence of odd perfect numbers. MASS selecta, 293–300, Amer. Math. Soc., Providence, RI, 2003.
  2. Kornel Dulęba. Perfect Numbers. slides. 2020.

(the seminar will only be online)

29.04.2020 12:15
Jakub Dyczek
Computer science foundations
On probabilistic term rewriting by Martin Avanzinia,Ugo Dal Lago and Akihisa Yamadac
We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method polynomial and matrix interpretations are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by means of multidistribution reduction sequences, thus accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.
23.04.2020 17:00
Bartłomiej Jachowicz
Combinatorial Optimization
Lonely runner conjecture

One of number theory open problem is the Lonely Runner Conjecture. It is interesting for several reasons. First the conjecture is relatively intuitive to grasp and easy to state. This conjecture can be find in two different contexts: as a problem in Diophantine’s approximation and as a geometric view obstruction problem. What is more, the difficulty of proving the Lonely Runner Conjecture may seem to increase exponentially with the number of runners. I present statement of the conjecture and known partial results.

  1. Clayton Barnes II. The Lonely Runner Conjecture. arXiv:1211.2482. 2012.
  2. Thomas W. Cusick, Jorg M. Wills. Lonely runner conjecture. Open Problem Garden. 2007.
  3. Bartłomiej Jachowicz. Lonely Runner Conjecture. slides. 2020.

(the seminar will only be online)

23.04.2020 16:15
Filip Bartodziej
Combinatorial Optimization
Meyniel’s conjecture on the cop number

A cops and robbers problem determines if the number of cops is sufficient to always catch a robber in a game with defined rules played on an undirected graph. Cop number of a graph is the minimal number of cops necessary for cops to win in that game on the specific graph. Mayniel’s conjuncture remains an open problem and states that cop number for graphs of order n is sqrt(n). I’ll present a survey of results achieved that are related to this conjecture.

  1. William Baird, Anthony Bonato. Meyniel's conjecture on the cop number: a survey. arXiv:1308.3385. 2013.
  2. Filip Bartodziej. Meyniel’s conjecture. slides. 2020.

(the seminar will only be online)

22.04.2020 12:15
Jan Kościsz
Computer science foundations
Fast Synchronization of Random Automata by Cyril Nicaud
A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerný conjectured in 1964 that if a n-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n − 1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^3 n) with high probability. As a consequence, this proves that most automata satisfy the Cerný conjecture.
16.04.2020 17:00
Mateusz Pabian
Combinatorial Optimization
Synchronizing Automata and the Černý Conjecture

I present many results and finally open problem related to synchronizing automata and synchronizing word sends any state of the DFA to one and the same state. This leads to the some natural problems such as: how can we restore control over such a device if we don't know its current state but can observe outputs produced by the device under various actions? I prove some uperbounds for length of this kind of word and in particular I will make a statement of Cerny conjecture.

  1. Mikhail V. Volkov. Synchronizing Automata and the Černý Conjecture. LATA 2008. 11-27.
  2. Mateusz Pabian. Synchronizing Automata and the Černý Conjecture. slides. 2020.

(the seminar will only be online)

16.04.2020 16:15
Adrian Siwiec
Combinatorial Optimization
Online Computation with Untrusted Advice

The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. We study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner.To this end, we focus on well-studied online problems such as ski rental, online bidding, bin packing, and list update.

  1. Spyros Angelopoulos, Christoph Durr, Shendan Jin, Shahin Kamali, and Marc Renault. Online Computation with Untrusted Advice. arXiv:1905.05655. 2019.
  2. Adrian Siwiec. Online Computation with Untrusted Advice. slides. 2020.

(the seminar will only be online)

15.04.2020 12:15
Magdalena Proszewska
Computer science foundations
Singular value automata and approximate minimization by Borja Balle, Prakash Panangaden and Doina Precup
The present paper uses spectral theory of linear operators to construct approximately minimal realizations of weighted languages. Our new contributions are: (i) a new algorithm for the singular value decomposition (SVD) decomposition of finite-rank infinite Hankel matrices based on their representation in terms of weighted automata, (ii) a new canonical form for weighted automata arising from the SVD of its corresponding Hankelmatrix, and (iii) an algorithm to construct approximate minimizations of given weighted automata by truncating the canonical form. We give bounds on the quality of our approximation.
09.04.2020 17:00
Wojciech Buczek
Combinatorial Optimization
Seymour's Second Neighbourhood Conjecture

Seymour's Second Neighbourhood Conjecture tells us, that any oriented graph has a vertex whose outdegree is at most its second outdegree, which is also known as Second neighborhood problem. Intuitively, it suggests that in a social network described by such a graph, someone will have at least as many friends-of-friends as friends. We will say about Chen-Shen-Yuster prove, that for any digraph D, there exists a vertex v such that |N++(v)|≥γ|N+(v)|, where γ=0.67815. We will consider graphs, in which we know, that such vertex exists. We will also say about unsuccessful attempts at proving this conjecture.

  1. Salman Ghazal. Seymour's second neighborhood conjecture for tournaments missing a generalized star. arXiv:1106.0085, 2011.
  2. Tyler Seacrest. Seymour's Second Neighborhood Conjecture for Subsets of Vertices. arXiv:1106.0085, 2018.
  3. Wojciech Buczek. Seymour’s Second Neighbourhood Conjecture. slides. 2020.

(the seminar will only be online)

09.04.2020 16:15
Mikołaj Twaróg
Combinatorial Optimization
Collatz conjecture

The Collatz conjecture, also known as 3n + 1 conjecture considers a function, which returns n/2 if n is even and 3n + 1 if n is odd. The conjecture states that for every n we can repeatedly apply this function to eventually reach 1. I will talk about different approaches to proving this conjecture.

  1. Mikołaj Twaróg. Collatz conjecture. slides. 2020.

(the seminar will only be online)

08.04.2020 12:15
Jacek Kurek
Computer science foundations
Complexity of translations from resolution to sequent calculus by GISELLE REIS and BRUNO PALEO
Resolution and sequent calculus are two well-known formal proof systems. Their differences make them suitable for distinct tasks. Resolution and its variants are very efficient for automated reasoning and are in fact the theoretical basis of many theorem provers. However, being intentionally machine oriented, the resolution calculus is not as natural for human beings and the input problem needs to be pre-processed to clause normal form. Sequent calculus, on the other hand, is a modular formalism that is useful for analysing meta-properties of various logics and is, therefore, popular among proof theorists. The input problem does not need to be pre-processed, and proofs are more detailed. However, proofs also tend to be larger and more verbose. When the worlds of proof theory and automated theorem proving meet, translations between resolution and sequent calculus are often necessary. In this paper, we compare three translation methods and analyse their complexity.
02.04.2020 17:00
Adam Pardyl
Combinatorial Optimization
Undirected edge geography

The game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex, erasing the edge in between. The player who first has no legal move loses the game. We analyze the space complexity of the decision problem of determining whether a start position in this game is a win for the first player. We also show a polynomial time algorithm for finding winning moves for bipartite graphs.

  1. Aviezri S. Fraenkel, Edward R. Scheinerman, Daniel Ullman. Undirected Edge Geography. Theoretical Computer Science112(2), 1993, 371-381.
  2. Adam Pardyl. Undirected edge geography. slides. 2020.

(the seminar will only be online)

02.04.2020 16:15
Piotr Mikołajczyk
Combinatorial Optimization
ARRIVAL game

Consider a directed graph such that every vertex has at most 2 outgoing edges - one of them is labeled as 'open' (we can traverse it) and the second one is labeled as 'closed' (we cannot traverse it). Every time we go somewhere from the vertex v, labels at its two edges are swapped, so the next time we visit v, we will take another direction. Given designated two vertices: origin and destination, we need to decide, whether eventually we will reach destination starting in the origin. This problem lies in both NP and coNP, but it is still an open question whether it belongs to P.

  1. Jérôme Dohrau, Bernd Gärtner, Manuel Kohler, Jiří Matoušek, Emo Welzl. ARRIVAL: A zero-player graph game in NP ∩ coNP. arXiv:1605.03546. 2020.
  2. Piotr Mikołajczyk. ARRIVAL game. slides. 2020.

(the seminar will only be online)

01.04.2020 12:15
Rafał Byczek
Computer science foundations
Bijection between oriented maps and weighted non-oriented maps by Agnieszka Czyzewska-Jankowska and Piotr Śniady
We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitrary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the
remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.
26.03.2020 16:15
Vladyslav Rachek
Combinatorial Optimization
Small weak epsilon-nets

Let P be a set of n points in R2. A point q (not necessarily in P) is called a centerpoint of P if each closed half-plane containing q at least ⌈n/3⌉ points of P, or, equivalently, any convex set that contains more than 2/n points of P must also contain q. It is a well-known fact that a centerpoint always exists and the constant 2/3 is the best possible. Can we improve this constant by using, say, two points, or some other small number of points? In the presentation we'll try to answer those questions.

Vladyslav Rachek. Small weak epsilon-nets. slides. 2020.

(the seminar will only be online)

 
25.03.2020 12:15
Michał Zwonek
Computer science foundations
FUNCTIONAL PEARL How to find a fake coin by RICHARD BIRD

The aim of this pearl is to solve the following well-known problem that appears in many puzzle books, for example Levitin & Levitin (2011) and Bellos (2016), usually for the particular case n=12.


There are n coins, all identical in appearance, one of which may be fake. The fake coin, if it exists, is either lighter or heavier than the fair coins, but it is not known which, nor by how much. Given is a balance with two pans but no weights. Equal number of coins can be placed on each pan and weighed. There are three possible outcomes: the left-hand pan may be lighter than the right-hand pan, or of equal weight, or heavier. Design a recipe for determining the minimum number of weighings guaranteed to determine the fake coin, if it exists, and whether it is lighter or heavier than the others.

19.03.2020 17:00
Kamil Rajtar
Combinatorial Optimization
How voting can be manipulated during selecting voting places

During today presentation we will learn how we can use graph theory to proof hardness of general problem of manipulating poll outcome. Based on paper: "Selecting Voting Locations for Fun and Profit" written by Zack Fitzsimmons and Omer Lev.

Zack Fitzsimmons, Omer Lev. Selecting Voting Locations for Fun and Profit. arXiv:2003.06879. 2020.

(the seminar will only be online)

19.03.2020 16:15
Mateusz Tokarz
Combinatorial Optimization
The Hadwiger-Nelson problem

We will focus on Hadwiger-Nelson problem - an open question from geometric graph theory that asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. There are a few interesting theorems related to the problem and results which we will go through. We will focus in particular on the most recent result of Aubrey de Grey who showed that the desired chromatic number is at least 5.

  1. Aubrey D.N.J. de Grey. The chromatic number of the plane is at least 5. arXiv:1804.02385. 2018.
  2. Mateusz Tokarz. slides. 2020.

(the seminar will only be online)

18.03.2020 12:15
Mateusz Tokarz, wyniki własne, kontynuacja
Computer science foundations
The largest fixed point in iterative programs
We study the smallest ordinal number α such that every Prologue program will reach its greatest fixed point after α downward iterations. Firstly, we show that the continuity of Prologue’s resolution function does not help with this matter. Then, due to the embedding of the recursive functions in Prologue, we get that α is at least Church-Kleene Omega. Using recursive linear order presented in “On the Forms of the Predicates in the Theory of Constructive Ordinals“ (Kleene, 1944) we construct a Prologue’s program requiring at least C-K Omega steps to achieve its greatest fixed point. To get the upper bound on α we use clockable ordinals introduced in “Infinite Time Turing Machines” (Joel David Hamkins, Andy Lewis, 1998).
11.03.2020 12:15
Mateusz Tokarz wyniki własne
Computer science foundations
The largest fixed point in iterative programs
We study the smallest ordinal number α such that every Prologue program will reach its greatest fixed point after α downward iterations. Firstly, we show that the continuity of Prologue’s resolution function does not help with this matter. Then, due to the embedding of the recursive functions in Prologue, we get that α is at least Church-Kleene Omega. Using recursive linear order presented in “On the Forms of the Predicates in the Theory of Constructive Ordinals“ (Kleene, 1944) we construct a Prologue’s program requiring at least C-K Omega steps to achieve its greatest fixed point. To get the upper bound on α we use clockable ordinals introduced in “Infinite Time Turing Machines” (Joel David Hamkins, Andy Lewis, 1998).
23.01.2020 16:15
Jan Gwinner
Combinatorial Optimization
Spectrally Robust Graph Isomorphism

In the paper authors consider certain variants of Graph Isomorphism problem. They focus on properties of graph spectra and eigenspaces - namely if Laplacian of one of the graphs is greater or equal to another in Loewner ordering. In the first part of the paper they prove that one of the problems named Spectral Graph Dominance is NPC. The rest of the paper is devoted to an approximation algorithm for special case of the problem called Spectrally Robust Graph Isomorphism.

Alexandra Kolla, Ioannis Koutis, Vivek Madan, Ali Kemal Sinop. Spectrally Robust Graph Isomorphism. arXiv:1805.00181, 1-17, 2018.

22.01.2020 12:15
Weronika Grzybowska i Mateusz Tokarz
Computer science foundations
On two subclasses of Motzkin paths and their relation to ternary trees by Helmut Prodinger, Sarah J. Selkirk and Stephan Wagner
Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.
16.01.2020 17:00
Gabriela Czarska
Combinatorial Optimization
Driver surge pricing

Authors study Uber's pricing mechanisms from the perspective of drivers, presenting the theoretical foundation that has informed the design of Uber’s new additive driver surge mechanism. They present a dynamic stochastic model to capture the impact of surge pricing on driver earnings and their strategies to maximize such earnings.

Nikhil Garg, Hamid Nazerzadeh. Driver Surge Pricing. arXiv. 2019.

16.01.2020 16:15
Bartosz Podkanowicz
Combinatorial Optimization
Planar graphs have bounded queue-number

The paper presents proof that the queue number of planar graphs is bounded. It also mentions generalizations of the result and other theorems that have similar proofs.

Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, David R. Wood. Planar graphs have bounded queue-number. arXiv. 2019.

16.01.2020 14:15
Katarzyna Król, Paweł Mader
Algorytmika
On the Complexity of Lattice Puzzles [Kobayashi et al.]
In this paper, authors investigate the computational complexity of lattice puzzle, which is one of the traditional puzzles. A lattice puzzle consists of 2n plates with some slits, and the goal of this puzzle is to assemble them to form a lattice of size n×n. It has a long history in the puzzle society; however, there is no known research from the viewpoint of theoretical computer science. This puzzle has some natural variants, and they characterize representative computational complexity classes in the class NP. Especially, one of the natural variants gives a characterization of the graph isomorphism problem. That is, the variant is GI-complete in general. As far as the authors know, this is the first non-trivial GI-complete problem characterized by a classic puzzle. Like the sliding block puzzles, this simple puzzle can be used to characterize several representative computational complexity classes.
15.01.2020 16:15
Michał Seweryn
Theoretical computer science
Erdös-Hajnal properties for powers of sparse graphs

The notion of nowhere dense classes of graphs attracted much attention in recent years and found many applications in structural graph theory and algorithmics. The powers of nowhere dense graphs do not need to be sparse, for instance the second power of star graphs are complete graphs. However, it is believed that powers of sparse graphs inherit somewhat simple structure. In this spirit, we show that for a fixed nowhere dense class of graphs, a positive real ε and a positive integer d, in any n-vertex graph G in the class, there are disjoint vertex subsets A and B with |A|=Ω(n) and |B|=Ω(n1-ε) such that in the d-th power of G, either there is no edge between A and B, or there are all possible edges between A and B.

 

 

Joint work with Marcin Briański, Piotr Micek and Michał Pilipczuk
15.01.2020 12:15
Wojciech Grabis
Computer science foundations
Ant colony optimization theory: A survey by Marco Dorigoa and Christian Blumb
Research on a new metaheuristic for optimization is often initially focused on proof-of-concept applications. It is only after experimental work has shown the practical interest of the method that researchers try to deepen their understanding of the method’s functioning not only through more and more sophisticated experiments but also by means of an effort to build a theory. Tackling questions such as “how and why the method works’’ is important, because finding an answer may help in improving its applicability. Ant colony optimization, which was introduced in the early 1990s as a novel technique for solving hard combinatorial optimization problems, finds itself currently at this point of its life cycle. With this article we provide a survey on theoretical results on ant colony optimization. First, we reviewsome convergence results. Then we discuss relations between ant colony optimization algorithms and other approximate methods for optimization. Finally, we focus on some research efforts directed at gaining a deeper understanding of the behavior of ant colony optimization algorithms. Throughout the paper we identify some open questions with a certain interest of being solved in the near future.
09.01.2020 17:00
Wojtek Grabis
Combinatorial Optimization
Algorithms for Destructive Shift Bribery.

Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods.

Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. arXiv. 2018.

09.01.2020 16:15
Dominik Gryboś
Combinatorial Optimization
Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice

The paper shows that the Diffie-Hellman protocol is not as secure as we thought. The authors present the Logjam attack, which consists in quickly calculating discrete logarithms based on previously performed calculations. This can be done because many websites use the same prime numbers in the message encryption process.

David Adrian, Karthikeyan Bhargavan, Zakir Durumeric, Pierrick Gaudry, Matthew Green, J. Alex Halderman, Nadia Heninger, Drew Springall, Emmanuel Thomé, Luke Valenta, Benjamin VanderSloot, Eric Wustrow, Santiago Zanella-Béguelink, Paul Zimmermann. Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice. WeakDH.org.

08.01.2020 12:15
Piotr Gaiński
Computer science foundations
How Similar Are Two Elections by Piotr Faliszewski, Piotr Skowron, Arkadii Slinko, Stanisław Szufa and Nimrod Talmon
We introduce the ELECTION ISOMORPHISM problem and a family of its approximate variants, which we refer to as d-ISOMORPHISM DISTANCE (d-ID) problems (where d is a metric between preference orders). We show that ELECTION ISOMORPHISM is polynomial-time solvable, and that the d-ISOMORPHISM DISTANCE problems generalize various classic rank-aggregation methods (e.g., those of Kemeny and Litvak). We establish the complexity of our problems (including their inapproximability) and provide initial experiments regarding the ability to solve them in practice.
19.12.2019 17:00
Kamil Kropiewnicki
Combinatorial Optimization
Impossibility of Distributed Consensus with One Faulty Proces

 

he consensus problem involves an asynchronous system of processes, some of which may be unreliable. The problem is for 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.

Authors of the paper were awarded a Dijkstra Prize for this work - given to the most influential papers in distributed computing.

Michael J. Fischer, Nancy A. Lynch, Michael S. Paterson. Impossibility of Distributed Consensus with One Faulty Process. Journal of the Association for Computing Machinery, 32(2), 1985, 374-382.

19.12.2019 16:15
Filip Bartodziej
Combinatorial Optimization
How to eat 4/9 of a pizza

Unevenly cut pizza is a frustrating occurrence. How can we then make sure that a friend is not trying to reduce our portion of the delicious meal? We will present a strategy which guarantees that one will leave the table satisfied, assuming that they started eating first.

Kolja Knauer, Piotr Micek, Torsten Ueckerdt. How to eat 4/9 of a pizza. arXiv. 2008.

18.12.2019 12:15
Bartosz Podkanowicz
Computer science foundations
Riordan arrays and combinatorial sums by Renzo Sprugnoli
The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems.
12.12.2019 17:00
Krzysztof Michalik
Combinatorial Optimization
Coloring planar graphs with 3 colors and no large monochromatic components

I will present a proof that there exists a function f(d), such that there exists a 3-coloring of any planar graph G in which each monochromatic subgraph has at most f(d) vertices, where d is the degree of the highest-degree vertex in G.

Louis Esperet, Gwenaël Joret. Coloring planar graphs with three colors and no large monochromatic components. arXiv, 1303.2487. 2013.

12.12.2019 16:15
Mateusz Kaczmarek
Combinatorial Optimization
Hadwiger’s conjecture

Survey of Hadwiger's Conjecture from 1943, that for all t ≥ 0, every graph is either t-colorable or has a subgraph that can be contracted to the complete t+1 vertices graph. This conjecture is the tremendous strengthening of the four-color problem also known as map coloring problem.

Paul Seymour. Hadwiger’s conjecture.

12.12.2019 14:15
Krzysztof Pióro, Krzysztof Potępa
Algorytmika
Linear-Space Data Structures for Range Mode Query in Arrays [Chan, Durocher, Larsen, Morrison, Wilkinson]
A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i,j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O(sqrt(n)*log log n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O(sqrt(n/log n)) worst-case time. Furthermore, we present strong evidence that a query time significantly below sqrt(n) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two sqrt(n) x sqrt(n) matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogonal range mode in higher dimensions (queries in near O(n1−1/2d) time) and for halfspace range mode in higher dimensions (queries in O(n1−1/d^2) time).
11.12.2019 16:15
Adam Polak
Theoretical computer science
Monochromatic triangles, intermediate matrix products, and convolutions

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(nω) time algorithms, for ω<2.373. On the other hand, the (min,+)-product, which is at the heart of APSP, is widely conjectured to require cubic time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems, e.g. the (min,max)-product, the min-witness product, APSP in directed unweighted graphs. The best runtimes for these "intermediate" problems are all O(n(3+ω)/2). A similar phenomenon occurs for convolution problems.


Can one improve upon the running times for these intermediate problems? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way?

We make progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and min-witness product can be reduced to both the (min,max)-product and a variant of the monochromatic triangle problem. We also show that a natural convolution variant of monochromatic triangle is equivalent to the famous 3SUM problem. As this variant is in O(n3/2) time and 3SUM is in O(n2) time, our result gives the first fine-grained equivalence between natural problems of different running times.

Joint work with Andrea Lincoln and Virginia Vassilevska Williams.

11.12.2019 12:15
Mateusz Górski
Computer science foundations
A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic by Paul Wild, Lutz Schroder, Dirk Pattinson and Barbara Konig.
The fuzzy modality probably is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that every formula in probabilistic fuzzy first-order logic that is non-expansive wrt. behavioural distance can be approximated by concepts of bounded rank in probabilistic fuzzy description logic.
05.12.2019 16:15
Kornel Dulęba
Combinatorial Optimization
The Return of Coppersmith’s Atack: Practical Factorization of Widely Used RSA Moduli

During the seminar I will discuss a clever attack on RSA library used in Infineon chips. Researchers discovered that the prime factors used for constructing private keys have a peculiar form. This allowed them to use a modified version of Coppersmith algorithm to recover private key basing on their public counterpart in a reasonable time for keys up to 2048 bit long.

Dusan Klinec, Vashek Matyas, Matus Nemec, Marek Sys, Petr Svenda. The Return of Coppersmith’s A‚ack: Practical Factorization of Widely Used RSA Moduli.

04.12.2019 12:15
Michał Zwonek
Computer science foundations
Probably Half True: Probabilistic Satisfability over Lukasiewicz Infnitely-valued Logic by Marcelo Finger and Sandro Preto
We study probabilistic-logic reasoning in a context that allows for "partial truths", focusing on computational and algorithmic properties of non-classical Lukasiewicz In nitely-valued Probabilistic Logic. In particular, we study the satis ability of joint probabilistic assignments, which we call LIPSAT. Although the search space is initially in nite, we provide linear algebraic methods that guarantee polynomial  size witnesses, placing LIPSAT complexity in the NP-complete class. An exact satis ability decision algorithm is presented which employs, as a subroutine, the decision problem for Lukasiewicz In nitely-valued (non probabilistic) logic, that is also an NP-complete problem. We develop implementations of the algorithms described and discuss the empirical presence of a phase transition behavior for those implementations.
28.11.2019 16:15
Mikołaj Twaróg
Combinatorial Optimization
A Short Guide to Hackenbush

Hackenbush is a two player game played on a graph with a few marked vertices. Players alternate turns. Each turn consists of removing one edge from the graph and all vertices that lost their connection to all marked ones. Player, that can't make a move, loses. I will present three different variants of Hackenbush(Red-Blue Hackenbush, Green Hackenbush and R-G-B Hackenbush) with methods to determine, which player has a winning strategy.

Padraic Bartlett. A Short Guide to Hackenbush. VIGRE REU 2006.

28.11.2019 14:15
Katarzyna Bułat, Dawid Tracz
Algorytmika
Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time [P. Parys]

Gry parzystości to gry pomiędzy dwoma graczami (zwyczajowo Even oraz Odd) na grafie skierowanym G = (V, E). Gracze przesuwają między wierzchołkami wirtualny token, tworząc ścieżkę. Wierzchołki grafu są etykietowane liczbami naturalnymi i każdy z nich jest przypisany do jednego gracza, który decyduje w jakim kierunku zostanie wykonany ruch z tego wierzchołka. Celem każdego z graczy jest wybranie takiej strategii, że przy nieskończonej grze (ścieżce), najwyższa nieskończenie wiele razy powtarzająca się etykieta będzie odpowiednio parzysta bądź nieparzysta.

Problem gry parzystości jest deterministyczny, to znaczy dla każdego wierzchołka jeden z graczy posiada strategię wygrywającą. Rekurencyjny algorytm Zielonki rozwiązuje grę parzystości w czasie wykładniczym. Istnieje jednak algorytm działający w czasie quasi-wielomianowym, czyli 2O((log(n))^c) dla pewnego, ustalonego c.

W trakcie prezentacji omówiony zostanie schemat nowej wersji algorytmu, przeprowadzona analiza jego złożoności oraz przedstawiony dowód poprawności zwracanego przez niego wyniku.

27.11.2019 16:15
20.11.2019 16:15
Patryk Mikos
Theoretical computer science
Efficient enumeration of non-isomorphic interval graphs

Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on n vertices with an O(n4) time delay between the output of two consecutive graphs. We improve their algorithm and achieve O(n3 log n) time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all n up to 15 (previous best was 12).

27.11.2019 12:15
Piotr Mikołajczyk
Computer science foundations
Satisfiability in Strategy Logic can be Easier than Model Checking by Erman Acar, Massimo Benerecetti and Fabio Mogavero.

In the design of complex systems, model-checking and satisfiability arise as two prominent decision problems. While
model-checking requires the designed system to be provided in advance, satisfiability allows to check if such a system even exists. With very few exceptions, the second problem turns out to be harder than the first one from a complexity-theoretic standpoint. In this paper, we investigate the connection between the two problems for a non-trivial fragment of Strategy Logic (SL, for short). SL extends LTL with first-order quantifications over strategies, thus allowing to explicitly reason about the strategic abilities of agents in a multi-agent system. Satisfiability for the full logic is known to be highly undecidable, while model-checking is non-elementary.

The SL fragment we consider is obtained by preventing strategic quantifications within the scope of temporal operators. The resulting logic is quite powerful, still allowing to express important game-theoretic properties of multi-agent systems, such as existence of Nash and immune equilibria, as well as to formalize the rational synthesis problem. We show that satisfiability for such a fragment is PSPACE-COMPLETE, while its model-checking complexity is 2EXPTIME-HARD. The result is obtained by means of an elegant encoding of the problem into the satisfiability of conjunctive-binding first-order logic, a recently discovered decidable fragment of first-order logic.

21.11.2019 17:00
Adrian Siwiec
Combinatorial Optimization
Edge Coloring Signed Graphs

We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. It then turns out that Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.

Richard Behr. Edge Coloring Signed Graphs. arXiv. 2018.

21.11.2019 16:15
Paweł Palenica
Combinatorial Optimization
Guess the Larger Number

We will discuss variations of a zero sum game where one player writes down two numbers on cards. The second player learns one of the numbers to make a guess which of the numbers is larger. We will show variations of the game where the second player has a greater chance of winning than 1/2.

Alexander Gnedin. Guess the Larger Number. arXiv. 2016.

21.11.2019 14:15
Jędrzej Kula, Przemysław Simajchel
Algorytmika
Subtree Isomorphism Revisited [A. Abboud et al.]

The Subtree Isomorphism problem asks whether a given tree is contained in an another given tree. The problem is of fundamental importance and has been studied since the 1960s. Subquadratic algorithms are known for some variants, e.g. ordered trees, but not in the general case.

We will present a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the SETH. In addition, we will show that the same lower bound holds even for the case of rooted trees of logarithmic height. To contrast the lower bound, we will also show subquadratic randomized algorithms for rooted trees of constant degree and logarithmic height.

The reductions utilize the new "tree gadgets" that are likely useful for future SETH-based lower bounds for problems on trees. The algorithms apply a folklore result from randomized decision tree complexity.

20.11.2019 12:15
Edyta Garbarz
Computer science foundations
Unifying Logical and Statistical AI Pedro by Domingos, Daniel Lowd, Stanley Kok, Aniruddh Nath, Hoifung Poon Matthew Richardson and Parag Singla
Intelligent agents must be able to handle the complexity and uncertainty of the real world. Logical AI has focused mainly on the former, and statistical AI on the latter. Markov logic combines the two by attaching weights to first-order formulas and viewing them as templates for features of Markov networks. Inference algorithms for Markov logic draw on ideas from satisfiability, Markov chain Monte Carlo and knowledge-based model construction. Learning algorithms are based on the voted perceptron, pseudo-likelihood and inductive logic programming. Markov logic has been successfully applied to a wide variety of problems in natural language understanding, vision, computational biology, social networks and others, and is the basis of the open-source Alchemy system.
13.11.2019 16:15
Grzegorz Guśpiel
Theoretical computer science
Smaller Universal Targets for Homomorphisms of Edge-Colored Graphs

The density D(G) of a graph G is the maximum ratio of the number of edges to the number of vertices ranging over all subgraphs of G. For a class F of graphs, the value D(F) is the supremum of densities of graphs in F.

A k-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,...,k. A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class F of graphs, a k-edge-colored graph H (not necessarily with the underlying graph in F) is k-universal for F when any k-edge-colored graph with the underlying graph in F admits a homomorphism to H. Such graphs are known to exist exactly for classes F of graphs with acyclic chromatic number bounded by a constant. The minimum number of vertices in a k-uniform graph for a class F is known to be Ω(kD(F)) and O(kd), where d is the ceiling of D(F) (result obtained in 2015 with Gutowski), and has been conjectured to be ϴ(kD(F)).

In this talk, I will present a construction of a k-universal graph on O(kd) vertices for any rational bound d on the density D(F). It follows that if D(F) is rational, the minimum number of vertices in a k-universal graph for F is indeed ϴ(kD(F)).

13.11.2019 12:15
Jan Kościsz
Computer science foundations
Bohm's Theorem, Church's Delta, Numeral Systems, and Ershov Morphisms by Richard Statman and Henk Barendregt
In this note we work with untyped lambda terms under beta-conversion and consider the possibility of extending Bohm's theorem to infnite RE (recursively enumerable) sets. Bohm's theorem fails in general for such sets V even if it holds for all fnite subsets of it. It turns out that generalizing Bohm's theorem to infnite sets involves three other superfcially unrelated notions; namely, Church's delta, numeral systems, and Ershov morphisms. Our principal result is that Bohm's theorem holds for an infnite RE set V closed under beta conversion iff V can be endowed with the structure of a numeral system with predecessor iff there is a Church delta (conditional) for V iff every Ershov morphism with domain V can be represented by a lambda term
07.11.2019 16:15
Kamil Rajtar
Combinatorial Optimization
A Price-Based Iterative Double Auction for Charger Sharing Markets

Jie Gao, Terrence Wong, Chun Wang, and Jia Yuan Yu. A Price-Based Iterative Double Auction for Charger Sharing Markets. arXiv. 2019.

07.10.2019 14:15
Nicoll Bryła, Mateusz Pabian
Algorytmika
Faster Algorithms for All Pairs Non-Decreasing Paths Problem [Duan, Jin, Wu]
In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time Õ(n^((3+ω)/2)) = Õ(n^2,686). Here n is the number of vertices, and ω < 2.373 is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for (max, min)-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for (max, min)-matrix product. The previous best upper bound for APNP on weighted digraphs was Õ(n^(1/2(3+(3-ω)/(ω+1) + ω))) = Õ(n^2,78) [Duan, Gu, Zhang 2018]. We also show an Õ(n^2) time algorithm for APNP in undirected graphs which also reaches optimal within logarithmic factors.
06.11.2019 12:15
Rafał Burczyński
Computer science foundations
Compaction of Church Numerals by Isamu Furuya and Takuya Kida
In this study, we address the problem of compaction of Church numerals. Church numerals are unary representations of natural numbers on the scheme of lambda terms. We propose a novel decomposition scheme from a given natural number into an arithmetic expression using tetration, which enables us to obtain a compact representation of lambda terms that leads to the Church numeral of the natural number. For natural number n, we prove that the size of the lambda term obtained by the proposed method is O((s log2n)^(log n/ (log log n))). Moreover, we experimentally confirmed that the proposed method outperforms binary representation of Church numerals on average, when n is less than approximately 10,000 .
30.10.2019 16:15
Bartosz Walczak
Theoretical computer science
Coloring and Maximum Weight Independent Set of rectangles

We prove that every intersection graph of axis-parallel rectangles in the plane with clique number ω has chromatic number O(ω log ω), which is the first improvement of the original O(ω2) bound of Asplund and Grünbaum from 1960. As a consequence, we obtain a polynomial-time O(log log n)-approximation algorithm for Maximum Weight Independent Set in axis-parallel rectangles, improving the previous best approximation ratio of O(log n/log log n).

Joint work with Parinya Chalermsook.

30.10.2019 12:15
Jan Mełech
Computer science foundations
On compressing and indexing repetitive sequences by Sebastian Kreft and Gonzalo Navarro
We introduce LZ-End, a new member of the Lempel–Ziv family of text compressors, which achieves compression ratios close to those of LZ77 but is much faster at extracting arbitrary text substrings. We then build the first self-index based on LZ77 (or LZ-End) compression, which in addition to text extraction offers fast indexed searches on the compressed text. This self-index is particularly effective for representing highly repetitive sequence collections, which arise for example when storing versioned documents, software repositories, periodic publications, and biological sequence databases.
24.10.2019 16:15
Vladyslav Rachek
Combinatorial Optimization
On Chromatic Number of Exact Distance Graphs

For any graph G and positive integer p we can consider "exact distance graph" G in which vertices x and y are connected if and only if their distance in G is exactly p. We can bound chromatic number of such graphs using notion of weak coloring numbers. Proof becomes particularly valuable for odd p and planar graphs G.

Jan van den Heuvel, H.A. Kierstead, Daniel A. Quiroz. Chromatic Numbers of Exact Distance Graphs. arXiv, abs/1612.02160. 2016.

23.10.2019 16:15
Gwenaël Joret
Université Libre de Bruxelles
Theoretical computer science
A new proof of the Erdős-Pósa theorem with applications

A classic result of Erdős and Pósa (1965) states that for every graph G and every integer k, either G has k vertex-disjoint cycles, or G has a set of O(k log k) vertices meeting all cycles. The usual way of proving this theorem is through the so-called frame technique. In this talk I will describe another equally simple way of proving the theorem, using a ball packing argument. Then I will describe some applications of this method, including to the variant where only cycles of length at least l are considered.

Joint work with Henning Bruhn, Wouter Cames van Batenburg, and Arthur Ulmer.

23.10.2019 12:15
Rafał Byczek
Computer science foundations
Suffix array and Lyndon factorization of a text by Sabrina Mantaci, Antonio Restivo, Giovanna Rosone and Marinella Sciortino
The main goal ofthis paper is to highlight the relationship between the suffix array of a text and its Lyndon factorization. It is proved in [15]that one can obtain the Lyndon factorization of a text from its suffix array. Conversely, here we show a new method for constructing the suffix array of a text that takes advantage of its Lyndon factorization. The surprising consequence of our results is that, in order to construct the suffix array, the local suffixes inside each Lyndon factor can be separately processed, allowing different implementative scenarios, such as online, external and internal memory, or parallel implementations. Based on our results, the algorithm that we propose sorts the suffixes by starting from the leftmost Lyndon factors, even if the whole text or the complete Lyndon factorization are not yet available.
17.10.2019 16:15
Vladyslav Rachek
Combinatorial Optimization
Steinberg's conjecture is false

It's commonly known that planar graphs are at most 4-colorable. One possible direction towards determining when planar graphs can be 3-colorable is to narrow to particular planar graphs with enforced structure. For example, one can forbid cycles of length 4,5,...,k where k>=4. There is a conjecture of Steinberg from 1976, that planar graphs without cycles of length 4 and 5 (as subgraphs) are 3-colorable. It has been open problem till 2016, when it was disproved in joint paper of Vincent Cohen-Addad, Michael Hebdige, Daniel Kral, Zhentao Li, Esteban Salgado, and we present proof from that paper.

Steinberg's Conjecture is false. Vincent Cohen-Addad, Michael Hebdige, Daniel Kral, Zhentao Li, Esteban Salgado. arXiv, abs/1604.05108. 2016.