### Working paper

## A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

An algorithm is implemented in the article for finding the independence number of a n-vertex graph from the class Free({P5,C5, Kp}) in time O(np+O(1)).

Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determining provability of bounded depth formulas in L* , the Lambek calculus with empty antecedents allowed. Pentus’ algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability in Lb* , the Lambek calculus with brackets allowing empty antecedents. Our algorithm runs in polynomial time when both the formula depth and the bracket nesting depth are bounded. It combines a Pentus-style tabularisation of proof nets with an automata-theoretic treatment of bracketing.

In this research it was considered the particular case of a railway problem, specifically, the construction of orders delivery schedule for one locomotive plying among three railway station. In this paper it was suggested a polynomial algorithm and were shown the results of a computing experiment.

In this paper we consider choice problems under the assumption that the preferences of the decision maker are expressed in the form of a parametric partial weak order without assuming the existence of any value function. We investigate both the sensitivity (stability) of each non-dominated solution with respect to the changes of parameters of this order, and the sensitivity of the set of non-dominated solutions as a whole to similar changes. We show that this type of sensitivity analysis can be performed by employing techniques of linear programming.

The paper examines the structure, governance, and balance sheets of state-controlled banks in Russia, which accounted for over 55 percent of the total assets in the country's banking system in early 2012. The author offers a credible estimate of the size of the country's state banking sector by including banks that are indirectly owned by public organizations. Contrary to some predictions based on the theoretical literature on economic transition, he explains the relatively high profitability and efficiency of Russian state-controlled banks by pointing to their competitive position in such functions as acquisition and disposal of assets on behalf of the government. Also suggested in the paper is a different way of looking at market concentration in Russia (by consolidating the market shares of core state-controlled banks), which produces a picture of a more concentrated market than officially reported. Lastly, one of the author's interesting conclusions is that China provides a better benchmark than the formerly centrally planned economies of Central and Eastern Europe by which to assess the viability of state ownership of banks in Russia and to evaluate the country's banking sector.