# Publications

This paper describes *BBMCW*, a new efficient exact maximum clique algorithm tailored for large sparse graphs which can be bit-encoded directly into memory without a heavy performance penalty. These graphs occur in real-life problems when some form of locality may be exploited to reduce their scale. One such example is correspondence graphs derived from data association problems. The new algorithm is based on the bit-parallel kernel used by the *BBMC* family of published exact algorithms. *BBMCW* employs a new bitstring encoding that we denote ‘watched’, because it is reminiscent of the ‘watched literal’ technique used in satisfiability and other constraint problems. The new encoding reduces the number of spurious operations computed by the *BBMC* bit-parallel kernel in large sparse graphs. Moreover, *BBMCW* also improves on bound computation proposed in the literature for bit-parallel solvers. Experimental results show that the new algorithm performs better than prior algorithms over data sets of both real and synthetic sparse graphs. In the real data sets, the improvement in performance averages more than two orders of magnitude with respect to the state-of-the-art exact solver *IncMaxCLQ*.

In print

Flow variations over time generalize standard network flows by introducing an element of time. In contrast to the classical case of static flows, a flow over time in such a network specifies a flow rate entering an arc for each point in time. In this setting, the capacity of an arc limits the rate of flow into the arc at each point in time. Traditionally, flows over time are computed in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static network flows, its drawback is the enormous size of the time-expanded network. In this paper, we extend the results about the minimum flow problem to network flows (with n nodes and m arcs) in which the time-varying lower bounds can involve both the source and the sink nodes (as in Fathabadi et al.) and also one additional node other than the source and the sink nodes. It is shown that this problem for the set (Formula presented.) of time points can be solved by at most n minimum flow computations, by suitably extending the dynamic minimum flow algorithm and reoptimization techniques. The running time of the presented algorithm is (Formula presented.).

In this paper we examine the maintenance decision support in classification of the dangerous situations discovered by the monitoring system. This task is reduced to the contextual multi-armed bandit problem. We highlight small sample size problem appeared in this task due to the rather rare failures. The novel algorithm based on the nearest neighbor search is proposed. An experimental study is provided for several synthetic datasets with the situations described by either simple features or grayscale images. It is shown, that our algorithm outperforms the well-known contextual multi-armed methods with the Upper Confidence Bound and softmax random search strategies.

This paper addresses the problem of insufficient performance of statistical classification with the medium-sized database (thousands of classes). Each object is represented as a sequence of independent segments. Each segment is defined as a random sample of independent features with the distribution of multivariate exponential type. To increase the speed of the optimal Kullback-Leibler minimum information discrimination principle, we apply the clustering of the training set and an approximate nearest neighbor search of the input object in a set of cluster medoids. By using the asymptotic properties of the Kullback-Leibler divergence, we propose the maximal likelihood search procedure. In this method the medoid to check is selected from the cluster with the maximal joint density (likelihood) of the distances to the previously checked medoids. Experimental results in image recognition with artificially generated dataset and Essex facial database prove that the proposed approach is much more effective, than an exhaustive search and the known approximate nearest neighbor methods from FLANN and NonMetricSpace libraries.

A class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

The notions of boundary and minimal hard classes of graphs, united by the term “critical classes”, are useful tools for analysis of computational complexity of graph problems in the family of hereditary graph classes. In this family, boundary classes are known for several graph problems. In the paper, we consider critical graph classes in the families of strongly hereditary and minor closed graph classes. Prior to our study, there was the only one example of a graph problem for which boundary classes were completely described in the family of strongly hereditary classes. Moreover, no boundary classes were known for any graph problem in the family of minor closed classes. In this article, we present several complete descriptions of boundary classes for these two families and some classical graph problems. For the problem of 2-additive approximation of graph bandwidth, we find a boundary class in the family of minor closed classes. Critical classes are not known for this problem

in the other two families of graph classes.

In this paper we focus on two essential problems of maintenance decision support systems, namely, 1) detection of potential dangerous situation, and 2) classification of this situation in order to recommend an appropriate repair action. The former task is usually solved with the known statistical process control techniques. The latter problem can be reduced to the contextual multi-armed bandit problem. We propose a novel algorithm with Bayesian classification of abnormal situation and the softmax rule to explore the decision space. The dangerous situations are detected with the Shewhart control charts for the distances between the current and the normal situations. It is experimentally shown, that our algorithm is more accurate than the known contextual multi-armed methods with stochastic search strategies.

Given a graph, the Edge minimum sum-of-squares clustering problem requires finding *p* prototypes (cluster centres) by minimizing the sum of their squared distances from a set of vertices to their nearest prototype, where a prototype can be either a vertex or an inner point of an edge. In this paper we have implemented Variable neighborhood search based heuristic for solving it. We consider three different local search procedures, K-means, J-means, and a new I-means heuristic. Experimental results indicate that the implemented VNS-based heuristic produces the best known results in the literature.

An exhaustive search of all classes in pattern recognition methods cannot be implemented in real-time, if the database contains a large number of classes. In this paper we introduce a novel probabilistic approximate nearest-neighbor (NN) method. Despite the most of known fast approximate NN algorithms, our method is not heuristic. The joint probabilistic densities (likelihoods) of the distances to previously checked reference objects are estimated for each class. The next reference instance is selected from the class with the maximal likelihood. To deal with the quadratic memory requirement of this approach, we propose its modification, which processes the distances from all instances to a small set of pivots chosen with the farthest-first traversal. Experimental study in face recognition with the histograms of oriented gradients and the deep neural network-based image features shows that the proposed method is much faster than the known approximate NN algorithms for medium databases.

In print

Research into the market graph is attracting increasing attention in stock market analysis. One of the important problems connected with the market graph is its identification from observations. The standard way of identifying the market graph is to use a simple procedure based on statistical estimations of Pearson correlations between pairs of stocks. Recently a new class of statistical procedures for market graph identification was introduced and the optimality of these procedures in the Pearson correlation Gaussian network was proved. However, the procedures obtained have a high reliability only for Gaussian multivariate distributions of stock attributes. One of the ways to correct this problem is to consider different networks generated by different measures of pairwise similarity of stocks. A new and promising model in this context is the sign similarity network. In this paper the market graph identification problem in the sign similarity network is reviewed. A new class of statistical procedures for the market graph identification is introduced and the optimality of these procedures is proved. Numerical experiments reveal an essential difference in the quality between optimal procedures in sign similarity and Pearson correlation networks. In particular, it is observed that the quality of the optimal identification procedure in the sign similarity network is not sensitive to the assumptions on the distribution of stock attributes.

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.

A class of distribution free multiple decision statistical procedures is proposed for threshold graph identification in correlation networks. The decision procedures are based on simultaneous application of sign statistics. It is proved that single step, step down Holm and step up Hochberg statistical procedures for threshold graph identification are distribution free in sign similarity network in the class of elliptically contoured distributions. Moreover it is shown that these procedures can be adapted for distribution free threshold graph identification in Pearson correlation network.

In print

We completely determine the complexity status of the vertex 3-colorability problem for the problem restricted to all hereditary classes defined by at most 3 forbidden induced subgraphs each on at most 5 vertices. We also present a complexity dichotomy for the problem and the family of all hereditary classes defined by forbidding an induced *bull* and any set of induced subgraphs each on at most 5 vertices.

In print

We consider boolean linear programming formulations of the independent set, the vertex and the edge dominating set problems and prove their polynomial-time solvability for classes of graphs with (augmented) constraint matrices having bounded minors in the absolute value

In print

The tolerance of an element of a combinatorial optimization problem with respect to its optimal solution is the maximum change of the cost of the element while preserving the optimality of the given optimal solution and keeping all other input data unchanged. Tolerances play an important role in the design of exact and approximation algorithms, but the computation of tolerances requires additional computational time. In this paper, we concentrate on combinatorial optimization problems for which the computation of all tolerances and an optimal solution have almost the same computational complexity as of finding an optimal solution only. We summarize efficient computational methods for computing tolerances for these problems and determine their time complexity experimentally.

We show that the chromatic number of {P_5,K_p-e}-free graphs can be computed in polynomial time for each fixed p.

Additionally, we prove polynomial-time solvability of the weighted vertex coloring problem for {P_5,co(P_3+P_2)}-free graphs.

Model selection for Gaussian concentration graph is based on multiple testing of pairwise cinditional independence. In practical applications partial correlation tests are widely used. However it is not known whether partial correlation test is uniformly most powerful for pairwise conditional independence testing. This question is answered in the paper.Uniformly most powerful unbiased test of Neymann structure is obtained. It turns out that this test can be reduced to usual partial correlation test. It implies that partial correlation test is uniformly most powerful unbiased one.

We study the vertex coloring problem in classes of graphs defined by finitely many forbidden induced subgraphs. Of our special interest are the classes defined by forbidden induced subgraphs with at most 4 vertices. For all but three classes in this family we show either NP-completeness or polynomial-time solvability of the problem. For the remaining three classes we prove fixed-parameter tractability. Moreover, for two of them we give a 3/2 approximation polynomial algorithm.