KOMBINATORYKA ZADANIA I ODPOWIEDZI PDF

Jak pan wspomina studia? Oczekiwanie natychmiastowej odpowiedzi ucznia na zadane pytanie jest wbrew istocie matematyki. Lata Recenzentami byli matematycy. Tej satysfakcji nie mam. Stefan Turnau Doktor habilitowany w zakresie dydaktyki matematyki.

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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.

In , 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. This is joint work with K. Abdelmoumen, D. Bennis, and F. The square G 2 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.

Equivalently, the square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In , Gerd Wegner proved that the square of cubic planar graphs is 8-colorable. Wegner also gave some examples to illustrate that these upper bounds can be obtained. However several upper bounds in terms of maximum degree D have been proved as follows.

Moreover, conjecture is confirmed in the case of outerplanar graphs and graphs without K 4 minor. The presentation will be based on my master thesis. 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.

Sharareh Alipour and Amir Jafari showed various upper bounds for minimal cardinality of a,b -dominating set. These tools allowed them to obtain well-known bounds in a simpler way or new improved bounds in some special cases, including regular graphs. 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 , but the original 3-flow conjecture remains wide open.

These problems and the surrounding results will be presented during the seminar. If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? A number is perfect if it is equal to the sum of its divisors. So far only even perfect numbers have been found. 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.

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.

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. 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. 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. 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 consider graphs, in which we know, that such vertex exists.

We will also say about unsuccessful attempts at proving this conjecture. 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. 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.

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. 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. 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. During today presentation we will learn how we can use graph theory to proof hardness of general problem of manipulating poll outcome.

Zack Fitzsimmons, Omer Lev. Selecting Voting Locations for Fun and Profit. 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.

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. The rest of the paper is devoted to an approximation algorithm for special case of the problem called Spectrally Robust Graph Isomorphism.

Spectrally Robust Graph Isomorphism. 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. 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. Planar graphs have bounded queue-number. 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. Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. 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. 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. 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. Journal of the Association for Computing Machinery, 32 2 , , Unevenly cut pizza is a frustrating occurrence.

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