### Article

## Tropical semimodules of dimension two

In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules S in R^n having topological dimension two are studied, and it is shown that any such S has a finite weak dimension not exceeding n. For a fixed k, a polynomial time algorithm is constructed that decides whether S is contained in some tropical semimodule of weak dimension k or not. This result provides a solution of a problem that has been open for eight years.

The tropical arithmetic operations on R are defined as (a,b) -> min{a,b} and (a,b) -> a+b. We are interested in the concept of a semimodule, which is rather ill-behaved in tropical mathematics. In our paper we study the semimodules S in R^n having topological dimension two, and we show that any such S has always a finite weak dimension not exceeding n. For a fixed k, we construct a polynomial time algorithm deciding whether S is contained in some tropical semimodule of weak dimension k or not. The latter result provides a solution of a problem that has been open for eight years.

This book constitutes the proceedings of the 13th International Computer Science Symposium in Russia, CSR 2018, held in Moscow, Russia, in May 2018.

The 24 full papers presented together with 7 invited lectures were carefully reviewed and selected from 42 submissions. The papers cover a wide range of topics such as algorithms and data structures; combinatorial optimization; constraint solving; computational complexity; cryptography; combinatorics in computer science; formal languages and automata; algorithms for concurrent and distributed systems; networks; and proof theory and applications of logic to computer science.

We investigate regular realizability (RR) problems, which are the prob- lems of verifying whether intersection of a regular language – the input of the problem – and fixed language called filter is non-empty. In this pa- per we focus on the case of context-free filters. Algorithmic complexity of the RR problem is a very coarse measure of context-free languages com- plexity. This characteristic is compatible with rational dominance. We present examples of P-complete RR problems as well as examples of RR problems in the class NL. Also we discuss RR problems with context- free filters that might have intermediate complexity. Possible candidates are the languages with polynomially bounded rational indices.

We discuss the online teaching of Linear algebra using the Wolfram Research software product called web- Mathematica. The teaching is based on interactive electronic tutorials developed by the author. The tutorials provide distant students with the instruments of remote calculation and visualization of the calculation results. All this increases the chances for students to deepen the understanding of the basic principles of Linear algebra and acquire the skills of solving problems.

Boolean games are an expressive and natural formalism through which to investigate problems of strategic interaction in multiagent systems. Although they have been widely studied, almost all previous work on Nash equilibria in Boolean games has focused on the restricted setting of pure strategies. This is a shortcoming as finite games are guaranteed to have at least one equilibrium in mixed strategies, but many simple games fail to have pure strategy equilibria at all. We address this by showing that a natural decision problem about mixed equilibria: determining whether a Boolean game has a mixed strategy equilibrium that guarantees every player a given payoff, is NEXP-hard. Accordingly, the epsilon variety of the problem is NEXP-complete. The proof can be adapted to show coNEXP-hardness of a similar question: whether all Nash equilibria of a Boolean game guarantee every player at least the given payoff.

For a graph *G* and a positive integer *k*, a subset *C* of vertices of *G* is called a *k*-path vertex cover if *C* intersects all paths of *k* vertices in *G*. The cardinality of a minimum *k*-path vertex cover is denoted by *β_{**P_**k*}(*G*). For a graph *G* and a positive integer *k*, a subset *M* of pairwise vertex-disjoint paths of *k* vertices in *G* is called a *k*-path packing. The cardinality of a maximum *k*-path packing is denoted by *μ**_{P_**k*}(*G*). In this paper, we describe some graphs, having equal values of *β**_{P_**k}* and *μ**{P**_k}*, for *k*≥5, and present polynomial-time algorithms of finding a minimum *k*-path vertex cover and a maximum *k*-path packing in such graphs.

We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk o MAJk. We observe that the minimum value of k for which there exists a MAJk o MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk o MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3+o(1). We show a worst case lower bound: if a MAJk o MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19+o(1). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k = O(n2/3) can compute MAJn correctly on all inputs.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.