On the graph limit question of Vera T. S\'os

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. S\'os proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.Comment: 4 page

Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page

Independent sets and cuts in large-girth regular graphs

We present a local algorithm producing an independent set of expected size $0.44533n$ on large-girth 3-regular graphs and $0.40407n$ on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with $1.34105n$ edges on large-girth 3-regular graphs. These decrease the gaps between the best known upper and lower bounds from $0.0178$ to $0.01$, from $0.0242$ to $0.0123$ and from $0.0724$ to $0.0616$, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of $1 / 0.44533 \approx 2.24554$ and $1 / 0.40407 \approx 2.4748$ and fractional edge coloring number $1.5 / 1.34105 \approx 1.1185$. Our algorithms are applications of the technique introduced by Hoppen and Wormald

Random local algorithms

Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which can use any local statistics about the graph, and produces an almost optimal structure, then the same can be achieved by a random local algorithm using no statistics.Comment: 9 page

An undecidability result on limits of sparse graphs

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (G_i) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of G_i is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the case where the sequence (G_i) is replaced by a unimodular random graph.Comment: 6 page

Generalized solution for the Herman Protocol Conjecture

We have a cycle of $N$ nodes and there is a token on an odd number of nodes. At each step, each token independently moves to its clockwise neighbor or stays at its position with probability $\frac{1}{2}$. If two tokens arrive to the same node, then we remove both of them. The process ends when only one token remains. The question is that for a fixed $N$, which is the initial configuration that maximizes the expected number of steps $E(T)$. The Herman Protocol Conjecture says that the $3$-token configuration with distances $\lfloor\frac{N}{3}\rfloor$ and $\lceil\frac{N}{3}\rceil$ maximizes $E(T)$. We present a proof of this conjecture not only for $E(T)$ but also for $E\big(f(T)\big)$ for some function $f:\mathbb{N}\rightarrow\mathbb{R}^{+}$ which method applies for different generalizations of the problem

Efficient Teamwork

Our goal is to solve both problems of adverse selection and moral hazard for multi-agent projects. In our model, each selected agent can work according to his private "capability tree". This means a process involving hidden actions, hidden chance events and hidden costs in a dynamic manner, and providing contractible consequences which are affecting each other's working process and the outcome of the project. We will construct a mechanism that induces truthful revelation of the agents' capability trees and chance events and to follow the instructions about their hidden decisions. This enables the planner to select the optimal subset of agents and obtain the efficient joint execution. We will construct another mechanism that is collusion-resistant but implements an only approximately efficient outcome. The latter mechanism is widely applicable, and the major application details will be elaborated.Comment: 51 pages. It contains some colored figures on the first few pages, but these are readable in black and whit