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## The computational complexity of weighted vertex coloring for {P_5,K_{2,3},K_{2,3}^+}-free graphs

In this paper, we show that the weighted vertex coloring problem can be solved in polynomial on the sum of vertex weights time for {P_5, K_{2,3}, K_{2,3}^+}-free graphs. As a corollary, this fact implies polynomial-time solvability of the unweighted vertex coloring problem for {P_5, K_{2,3}, K_{2,3}^+}-free graphs. As usual, P_5 and K_{2,3} stands, respectively, for the simple path on 5 vertices and for the biclique with the parts of 2 and 3 vertices, K_{2,3} denotes the graph, obtained from a K_{2,3} by joining its degree 3 vertices with an edge.

This article is dedicated to an alternative method of solving of the Chinese Remainder Theorem for polynomials. To construct the solution, a system of linear equations is constructed (using the method of undetermined coefficients) and then solved. The complexity of the proposed method is also calculated.

This two-volume set (CCIS 905 and CCIS 906) constitutes the refereed proceedings of the Second International Conference on Advances in Computing and Data Sciences, ICACDS 2018, held in Dehradun, India, in April 2018. The 110 full papers were carefully reviewed and selected from 598 submissions. The papers are centered around topics like advanced computing, data sciences, distributed systems organizing principles, development frameworks and environments, software verification and validation, computational complexity and cryptography, machine learning theory, database theory, probabilistic representations.

We show that Branching-time temporal logics **CTL** and **CTL***, as well as Alternating-time temporal logics **ATL** and **ATL***, are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for **CTL**, as well as for **ATL**, with a single variable is EXPTIME-complete, while satisfiability for **CTL***, as well as for **ATL***, with a single variable is 2EXPTIME-complete,—i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiability for arbitrary formulas.

We consider the coloring problem for hereditary graph classes, i.e. classes of simple unlabeled graphs closed under deletion of vertices. For the family of the hereditary classes of graphs defined by forbidden induced subgraphs with at most four vertices, there are three classes with an open complexity of the problem. For the problem and the open three cases, we present approximation polynomial-time algorithms with performance guarantees.

Modal logics, both propositional and predicate, have been used in computer science since the late 1970s. One of the most important properties of modal logics of relevance to their applications in computer science is the complexity of their satisﬁability problem. The complexity of satisﬁability for modal logics is rather high: it ranges from NP-complete to undecidable for propositional logics and is undecidable for predicate logics. This has, for a long time, motivated research in drawing the borderline between tractable and intractable fragments of propositional modal logics as well as between decidable and undecidable fragments of predicate modal logics. In the present thesis, we investigate some very natural restrictions on the languages of propositional and predicate modal logics and show that placing those restrictions does not decrease complexity of satisﬁability. For propositional languages, we consider restricting the number of propositional variables allowed in the construction of formulas, while for predicate languages, we consider restricting the number of individual variables as well as the number and arity of predicate letters allowed in the construction of formulas. We develop original techniques, which build on and develop the techniques known from the literature, for proving that satisﬁability for a ﬁnite-variable fragment of a propositional modal logic is as computationally hard as satisﬁability for the logic in the full language and adapt those techniques to predicate modal logics and prove undecidability of fragments of such logics in the language with a ﬁnite number of unary predicate letters as well as restrictions on the number of individual variables. The thesis is based on four articles published or accepted for publication. They concern propositional dynamic logics, propositional branchingand alternating-time temporal logics, propositional logics of symmetric rela tions, and ﬁrst-order predicate modal and intuitionistic logics. In all cases, we identify the “minimal,” with regard to the criteria mentioned above, fragments whose satisﬁability is as computationally hard as satisﬁability for the entire logic.

We show that the weighted coloring problem can be solved for {P5,banner}-free graphs and for {P5,dart}-free graphs in polynomial time on the sum of vertex weights.

The task of complete complexity dichotomy is to clearly distinguish between easy and hard cases of a given problem on a family of subproblems. We consider this task for some optimization problems restricted to certain classes of graphs closed under deletion of vertices. A concept in the solution process is based on revealing the so-called “critical” graph classes, which play an important role in the complexity analysis for the family. Recent progress in studying such classes is presented in the article.

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

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.