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## Sprague-Grundy function of matroids and related hypergraphs

We consider a generalization of the classical game of Nim called hypergraph Nim. Given a hypergraph H on the ground set V={1,…,n} of *n* piles of stones, two players alternate in choosing a hyperedge H∈H and strictly decreasing all piles i∈H. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph Nim for several classes of hypergraphs. In particular we characterize all 2-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class.

We consider a generalization of the classical game of Nim called hypergraph Nim. Given a hypergraph H on the ground set V={1,…,n} of *n* piles of stones, two players alternate in choosing a hyperedge H∈H and strictly decreasing all piles i∈H. The player who makes the last move is the winner. In 1980 Jenkyns and Mayberry obtained an explicit formula for the Sprague–Grundy function of the hypergraph Nim whose hypergraph contains as the hyperedges all proper subsets of *V* (that is, all except ∅ and *V* itself). Somewhat surprisingly, the same formula works for a very wide family of hypergraphs. In this paper we characterize symmetric hypergraphs in this family.

Playing impartial games under the normal and misere conventions may differ a lot. However, there are also many "exceptions" for which the normal and misere Sprague-Grundy functions are very similar. The first such example, the game Nim, was considered by Bouton as early as in 1901. In 1976 Conway introduced a large class of such games that he called tame games. Here we introduce a proper subclass, pet games, and a proper superclass, domestic games. For each of these three classes we provide efficiently verifiable characterizations. These games are closely related to another important subclass of the tame games introduced in 2007 by the first author and called miserable games. We show that tame, pet, and domestic games turn into miserable games by "slight modifications" of the definitions. We also show that the sum of miserable games is miserable and find several other classes that respect summation. The developed techniques allow us to prove that very many well-known impartial games fall into classes mentioned above. Such examples include all subtraction games, which are pet; game Euclid, which is miserable (and, hence, tame), as well as many versions of the Wythoff game and Nim, which may be miserable, pet, or domestic.

In this talk I summarize the results obtained in 1999–2008 by Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassiony, Kazuhisa Makino, and myself, on complexity of generation algorithms. These algorithms can be partitioned into three groups: supergraph, flash-light (backtrack), and dual-bounded generation. We will call a problem *tractable* if it can be solved by a polynomial (nconstnconst) or quasi-polynomial (npolylog(n)npolylog(n)) time algorithm. More generally, for any positive non-decreasing function g=g(n)g=g(n), generating can be performed in total or incremental time *g*, or with *g*-delay. Most of the polynomial delay algorithms are provided by the flash-light (backtrack) method. As for the incremental algorithms, generating the next object is equivalent with just verifying its existence, which is a standard decision problem. Thus, incremental generation, in contrast to the delay one, may be NP-hard or NP-complete. For example, we show that generating all vertices of a polyhedron, given by its facets, is NP-complete (while the complexity status is still open in case of the polytopes, that is, bounded polyhedra). This problem is reduced to generating all negative cycles of a weighted digraph, which is NP-complete (for graphs, too). Generating all minimal transversals to a hypergraph, so-called dualization, plays an important role. For this problem an incremental quasi-polynomial algorithm (but no polynomial one) is known. We outline several wide classes of generation problems that can be reduced to dualization and, thus, solved in incremental quasi-polynomial time. We survey algorithms and complexity bounds for the above and many other generation problems.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

Existing approaches suggest that IT strategy should be a reflection of business strategy. However, actually organisations do not often follow business strategy even if it is formally declared. In these conditions, IT strategy can be viewed not as a plan, but as an organisational shared view on the role of information systems. This approach generally reflects only a top-down perspective of IT strategy. So, it can be supplemented by a strategic behaviour pattern (i.e., more or less standard response to a changes that is formed as result of previous experience) to implement bottom-up approach. Two components that can help to establish effective reaction regarding new initiatives in IT are proposed here: model of IT-related decision making, and efficiency measurement metric to estimate maturity of business processes and appropriate IT. Usage of proposed tools is demonstrated in practical cases.