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## The clique problem for graphs with a few eigenvalues of the same sign

The quadratic programming problem is known to be NP-hard for Hessian matrices with only one negative eigenvalue, but it is tractable for convex instances. These facts yield to consider the number of negative eigenvalues as a complexity measure

of quadratic programs. We prove here that the clique problem is tractable for two variants of its Motzkin-Strauss quadratic formulation with a fixed number of negative eigenvalues (with multiplicities).

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.

The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.

A simple measure of similarity for the construction of the market graph is proposed. The measure is based on the probability of the coincidence of the signs of the stock returns. This measure is robust, has a simple interpretation, is easy to calculate and can be used as measure of similarity between any number of random variables. For the case of pairwise similarity the connection of this measure with the sign correlation of Fechner is noted. The properties of the proposed measure of pairwise similarity in comparison with the classic Pearson correlation are studied. The simple measure of pairwise similarity is applied (in parallel with the classic correlation) for the study of Russian and Swedish market graphs. The new measure of similarity for more than two random variables is introduced and applied to the additional deeper analysis of Russian and Swedish markets. Some interesting phenomena for the cliques and independent sets of the obtained market graphs are observed.

In this chapter, we present our enhancements of one of the most efficient exact algorithms for the maximum clique problem—MCS algorithm by Tomita, Sutani, Higashi, Takahashi and Wakatsuki (in Proceedings of WALCOM’10, 2010, pp. 191–203). Our enhancements include: applying ILS heuristic by Andrade, Resende and Werneck (in Heuristics 18:525–547, 2012) to find a high-quality initial solution, fast detection of clique vertices in a set of candidates, better initial coloring, and avoiding dynamic memory allocation. A good initial solution considerably reduces the search tree size due to early pruning of branches related to small cliques. Fast detecting of clique vertices is based on coloring. Whenever a set of candidates contains a vertex adjacent to all candidates, we detect it immediately by its color and add it to the current clique avoiding unnecessary branching. Though dynamic memory allocation allows to minimize memory consumption of the program, it increases the total running time. Our computational experiments show that for dense graphs with a moderate number of vertices (like the majority of DIMACS graphs) it is more efficient to store vertices of a set of candidates and their colors on stack rather than in dynamic memory on all levels of recursion. Our algorithm solves p_hat1000-3 benchmark instance which cannot be solved by the original MCS algorithm. We got speedups of 7, 3000, and 13000 times for gen400_p0.9_55, gen400_p0.9_65, and gen400_p0.9_75 instances, correspondingly.

Many efficient exact branch and bound maximum clique solvers use approximate coloring to compute an upper bound on the clique number for every subproblem. This technique reasonably promises tight bounds on average, but never tighter than the chromatic number of the graph.

Li and Quan, 2010, AAAI Conference, p. 128–133 describe a way to compute even tighter bounds by reducing each colored subproblem to maximum satisfiability problem (MaxSAT). Moreover they show empirically that the new bounds obtained may be lower than the chromatic number.

Based on this idea this paper shows an efficient way to compute related “infra-chromatic” upper bounds without an explicit MaxSAT encoding. The reported results show some of the best times for a stand-alone computer over a number of instances from standard benchmarks.

This book constitutes the refereed proceedings of the 23rd Annual Symposium on Combinatorial Pattern Matching, CPM 2012, held in Helsinki, Finalnd, in July 2012. The 33 revised full papers presented together with 2 invited talks were carefully reviewed and selected from 60 submissions. The papers address issues of searching and matching strings and more complicated patterns such as trees, regular expressions, graphs, point sets, and arrays. The goal is to derive non-trivial combinatorial properties of such structures and to exploit these properties in order to either achieve superior performance for the corresponding computational problems or pinpoint conditions under which searches cannot be performed efficiently. The meeting also deals with problems in computational biology, data compression and data mining, coding, information retrieval, natural language processing, and pattern recognition.

Artifacts caused by intensely absorbing areas are encountered in computed tomography and may obscure or simulate pathology in medical applications, hide or mimic the cracks and cavities in the devices at industrial applications. We simulated sinograms with different levels of absorption to demonstrate the artifacts dynamics. If the analysis of the measured data shows the presence of strongly absorbing areas in the object under study we propose to use quadratic programming technique for solving the inverse problem. Although the technique is time-consuming it allows us to avoid the typical artifacts. We compare the images reconstructed with different techniques including the proposed one. ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors).

This book constitutes the refereed proceedings of the 44th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2018, held in Krems, Austria, in January/February 2018. The 48 papers presented in this volume were carefully reviewed and selected from 97 submissions. They were organized in topical sections named: foundations of computer science; software engineering: advances methods, applications, and tools; data, information and knowledge engineering; network science and parameterized complexity; model-based software engineering; computational models and complexity; software quality assurance and transformation; graph structure and computation; business processes, protocols, and mobile networks; mobile robots and server systems; automata, complexity, completeness; recognition and generation; optimization, probabilistic analysis, and sorting; filters, configurations, and picture encoding; machine learning; text searching algorithms; and data model engineering.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.