### Article

## Pareto-optimal Algorithms for Metric TSP: Experimental Research

The Travelling Salesman Problem (TSP) is a fundamental task in combinatorial optimization. A special case of the TSP is Metric TSP, where the triangle inequality holds. Solutions of the TSP are generally used for costs minimization, such as finding the best tour for round-the-world trip or construction of very large-scale integration schemes. Since the TSP is NP-hard, heuristic algorithms providing near optimal solutions will be considered. The objective of this article is to find a group of Pareto optimal heuristic algorithms for Metric TSP under criteria of run time efficiency and qualitative performance as a part of the experimental study. Classification of algorithms for Metric TSP is presented. Feasible heuristic algorithms and their prior estimates are described. The data structure and the details of the research methodology are provided. Finally, results and prospective research are discussed.

*Full text (PDF, 785 Kb)*

The objective of this research is to develop a software algorithm for evaluating the effectiveness of using the Financial Message Transfer System for Remote Banking Systems. The study considers hardware system that provide two types of interactions between the legal entity and the bank: ≪Direct Banking≫ and ≪Internet Banking≫. During the work were identified and analyzed the main features of remote banking systems and determined the efficiency criteria of the systems. Based on the analysis output, were developed and proposed a mathematical model for evaluating the effectiveness of considered systems thought using the cost and quantity coefficients of system ensuring, as well as the financial and economic business performance indicators. In accordance with the proposed mathematical model, was constructed an algorithm that makes it possible to obtain quantitative estimates of Remote Banking Systems effectiveness. The developed algorithm belongs to the heuristic class and is based on the analysis of both the open statistical and the modeled by simulation tools data. The conducted computational experiments reflects the efficiency of the proposed mathematical model, as well as an algorithm based on the developed model.

The classical Branch and Bound algorithm for the travelling salesman problem, presented in 1963 by Little J.D.C., Murty K.G., Sweeney D.W., Karel C., is nowadays one of the most popular algorithms to find a minimum Hamiltonian cycle in a complete graph. There are many literature references having an algorithm pseudocode with comments. However, there are some special cases which are not discussed in these references. One of these cases is an-alyzedin this article.

The article is devoted to a new generic representation of classes of particular travelling salesman problems (TSP), we call it "index number matrix". This representation is intended for a separation of problem classes of similar complicity. This result is destined to solve a problem of particular TSP’s complicity prediction. Two hypotheses are given regarding to an appliance of the representation, that is proposed in the article. A method of index number matrix construction is also given. Some results are obtained by experiments, which didn’t interfere with one of our hypotheses.

Motivated by applications in manufacturing industry, we consider a supply chain scheduling problem, where each job is characterised by non-identical sizes, different release times and unequal processing times. The objective is to minimise the makespan by making batching and sequencing decisions. The problem is formalised as a mixed integer programming model and proved to be strongly NP-hard. Some structural properties are presented for both the general case and a special case. Based on these properties, a lower bound is derived, and a novel two-phase heuristic (TP-H) is developed to solve the problem, which guarantees to obtain a worst case performance ratio of . Computational experiments with a set of different sizes of random instances are conducted to evaluate the proposed approach TP-H, which is superior to another two heuristics proposed in the literature. Furthermore, the experimental results indicate that TP-H can effectively and efficiently solve large-size problems in a reasonable time.

The routing problems are important for logistic and transport sphere. Basically, the routing problems related to determining the optimal set of routes in the multigraph. The Chinese postman problem (CPP) is a special case of the routing problem, which has many potential applications. We propose to solve the MCPP (special NP-hard case of CPP, which defined on mixed multigraph) using the reduction of the original problem into General Travelling Salesman Problem (GTSP). The variants of CPP are pointed out. The mathematical formulations of some problems are presented. The algorithm for reduction the MCPP in multigraph into GTSP is shown. The experimental results of solving MCPP in multigraph through the reduction into GTSP are presented.

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.

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXPNP, or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are *selfreducible*? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.