# Michael KhachayRussian Academy of Sciences | RAS · Krasovsky Institute of Mathematics and Mechanics

Michael Khachay

28.05

·

Professor

## About

158

Publications

11,522

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376

Citations

Introduction

Current research scope includes complexity, approximability issues, and algorithm design for intractable combinatorial optimization problems (e.g. generalisations of TSP and VRP, geometric settings of the Set Cover and Hitting Set problems, etc.). Also, I am involved in development of efficient ensemble machine learning techniques (committee machines)

Research Experience

September 2012 - present

**Omsk State Technical University**

Position

- Visiting Professor

July 2005 - present

**Ural Federal University**

Position

- Professor (Full)

Description

- Pattern Recognition, Statistical Learning, Mathematical Economics

July 2005 - present

**Ural Federal University**

Position

- Professor (Full)

Education

February 2005 - February 2005

September 1993 - December 1996

September 1988 - June 1993

## Publications

Publications (158)

We consider the Minimum Affine Separating Committee (MASC) combinatorial optimization problem, which is related to ensemble machine learning techniques on the class of linear weak classifiers combined by the rule of simple majority. Actually, the MASC problem is a mathematical formal- ization of the famous Vapnik-Chervonenkis principle of structura...

The cycle cover problem is a combinatorial optimization problem, which is to find a minimum cost cover of a given weighted digraph by a family of vertex-disjoint cycles. We consider a special case of this problem, where, for a fixed number k, all feasible cycle covers are restricted to be of the size k. We call this case the minimum weight k-size c...

The topological properties of a pseudometric space defined by a measure are investigated. Criteria of compactness and σ-compactness of this space are proved. A new sufficient condition for the uniform convergence (over an event class) of frequencies
to probabilities is proved as a corollary.
Keywordsmetric Boolean algebra–sigma-compactness–uniform...

Two special cases of the Minimum Committee Problem are studied, the Minimum Committee Problem of Finite Sets (MCFS) and the Minimum Committee Problem of a System of Linear Inequalities(MCLE). It is known that the flrst of these problems is NP-hard (see (1)). In this paper we show the NP-hardness of two integer optimization problems connected with i...

It is known that the minimum affine separating committee (MASC) combinatorial op- timization problem, which is related to some machine learning techniques, is NP-hard and does not belong to Apx class unless P = NP . In this paper, it is shown that the MASC problem for- mulated in a fixed dimension space within n> 1 is intractable even if sets defin...

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having numerous practically important applications. CVRP is strongly NP-hard (even on the Euclidean plane), hard to approximate in the general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there...

The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) is the well-known combinatorial optimization problem having numerous valuable applications in operations research. Unlike the classic CVRP (without time windows constraints), approximability of the CVRPTW (even in the Euclidean plane) in the class of algorithms with theoretical guar...

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem that has numerous valuable practical applications. It is known, that CVRP is strongly NP-hard even on the Euclidean plane and APX-hard in its metric setting even for any fixed capacity . For the Euclidean setting, there are known several approximatio...

This book constitutes the refereed proceedings of the 10th International Conference on Optimization and Applications, OPTIMA 2019, held in Petrovac, Montenegro, in September-October 2019.
The 35 revised full papers presented were carefully reviewed and selected from 117 submissions. The papers cover such topics as optimization, operations research...

This book constitutes the proceedings of the 8th International Conference on Analysis of Images, Social Networks and Texts, AIST 2019, held in Kazan, Russia, in July 2019.
The 24 full papers and 10 short papers were carefully reviewed and selected from 134 submissions (of which 21 papers were rejected without being reviewed). The papers are organiz...

This book constitutes the proceedings of the 19th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2020, held in Novosibirsk, Russia, in July 2020. The 31 full papers presented in this volume were carefully reviewed and selected from 102 submissions. The papers are grouped in these topical sections: discre...

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having numerous relevant applications in operations research. As known, CVRP is strongly NP-hard even in the Euclidean plane, APX-hard for an arbitrary metric, and can be approximated in polynomial time with any accuracy in the Euclidean spaces of an...

The capacitated vehicle routing problem with time windows (CVRPTW) is a well-known NP-hard combinatorial optimization problem. We present a further development of the approach first proposed by M. Haimovich and A. H. G. Rinnooy Kan and propose an algorithm that, for an arbitrary ε > 0, finds a (1 + ε)-approximate solution for the Euclidean CVRPTW i...

This volume contains the refereed proceedings of the 18th international conference on
Mathematical Optimization Theory and Operations Research (MOTOR 2019) held
during July 8–12, 2019, near Ekaterinburg, Russia.
The conference brings together a wide research community in the fields of
mathematical programming and global optimization, discrete optim...

The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) is the well-known combinatorial optimization problem having numerous valuable applications in operations research. Unlike the classic CVRP (without time windows constraints), approximation algorithms with theoretical guarantees for the CVRPTW are still developed much less, even for...

We consider the geometric version of the well-known Generalized Traveling Salesman Problem introduced in 2015 by Bhattacharya et al. that is called the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC). They proved the intractability of the problem and proposed first polynomial time algorithms with fixed approximation fac...

This book constitutes the refereed proceedings of the 9th International Conference on Optimization and Applications, OPTIMA 2018, held in Petrovac, Montenegro, in October 2018.
The 35 revised full papers and the one short paper presented were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections on mathe...

This research is motivated by sustainability problems of oil palm expansion. Fast-growing industrial Oil Palm Plantations (OPPs) in the tropical belt of Africa, Southeast Asia and parts of Brazil lead to significant loss of rainforest and contribute to the global warming by the corresponding decrease of carbon dioxide absorption. We propose a novel...

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having a wide range of practical applications in operations research. It is known that the problem is NP-hard and remains intractable even in the Euclidean plane. Although the problem is hardly approximable in the general case, some of its geometric...

This research is motivated by the global warming problem, which is likely influenced by human activity. Fast-growing oil palm plantations in the tropical belt of Africa, Southeast Asia and parts of Brazil lead to significant loss of rainforest and contribute to the global warming by the corresponding decrease of carbon dioxide absorption. We propos...

Generalized Traveling Salesman Problem (GTSP) is a well-known combinatorial optimization problem having numerous applications in operations research. For a given edge-weighted graph and a partition of its nodeset onto k (disjoint) clusters it is required to find a minimum cost cyclic tour visiting all the clusters once. The problem is strongly NP-h...

This book constitutes the proceedings of the 7th International Conference on Analysis of Images, Social Networks and Texts, AIST 2018, held in Moscow, Russia, in July 2018.
The 29 full papers were carefully reviewed and selected from 107 submissions (of which 26 papers were rejected without being reviewed). The papers are organized in topical sect...

We consider the famous k-medians clustering problem in the context of a zero-sum two-player game, which is defined as follows. For given integers \(n>1\) and \(k>1\), strategy sets of the first and second players consist of n-samples drawn from the unit segment [0, 1] and partitions of the index set \(\{1,\ldots , n\}\) into k nonempty subsets (clu...

This book constitutes extended, revised and selected papers from the 7th International Conference on Optimization Problems and Their Applications, OPTA 2018, held in Omsk, Russia in July 2018. The 27 papers presented in this volume were carefully reviewed and selected from a total of 73 submissions. The papers are listed in thematic sections, namel...

We consider the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC), a special geometric subclass of the famous Generalized TSP, introduced by Bhattacharya et al. They showed that the problem is strongly NP-hard if the number of clusters k belongs to the instance and proposed the first polynomial time algorithm with a fixed...

We describe the possibility of employing the special case of the 3-SAT problem stemming from the well known integer factorization problem for the quantum cryptography. It is known, that for every instance of our 3-SAT setting the given 3-CNF is satisfiable by a unique truth assignment, and the goal is to find this assignment. Since the complexity s...

The Generalized Traveling Salesman Problem on Grid Clusters (GTSP-GC) is the geometric setting of the famous Generalized Traveling Salesman Problem, where the nodes of a given graph are points on the Euclidean plane and the clusters are defined implicitly by the cells of a unit grid. The problem in question is strongly NP-hard but can be approximat...

The Hitting Set Problem is the well known discrete optimization problem adopting interest of numerous scholars in graph theory, computational geometry, operations research, and machine learning. The problem is NP-hard and remains intractable even in very specific settings, e.g., for axis-parallel rectangles on the plane. Recently, for unit squares...

This book constitutes the proceedings of the 6th International Conference on Analysis of Images, Social Networks and Texts, AIST 2017, held in Moscow, Russia, in July 2017.
The 29 full papers and 8 short papers were carefully reviewed and selected from 127 submissions. The papers are organized in topical sections on natural language processing; gen...

In this paper, we introduce notions of l-quasi-pyramidal and l-pseudo-pyramidal tours extending the classic notion of pyramidal tour to the case of Generalized Traveling Salesman Problem (GTSP). We show that, for the instance of GTSP on n cities and k clusters with arbitrary weights, optimal l-quasi-pyramidal and l-pseudo-pyramidal tours can be fou...

In this paper, one-dimensional k-medians clustering problem is considered in the context of zero-sum game between players choosing a sample and partitioning it into clusters, respectively. For any sample size n and k > 1, an attainable guaranteed value of the clustering accuracy 0.5n/(2k − 1) (the low value of an appropriate game) is provided for s...

The Generalized Traveling Salesman Problem (GTSP) is defined by a weighted graph G = (V,E,w) and a partition of its vertex set into k disjoint clusters V = V1 ∪.. ∪ Vk. It is required to find a minimum-weight cycle that contains exactly one vertex of each cluster. We consider a geometric setting of the problem (we call it the EGTSP-k-GC), in which...

The Capacitated Vehicle Routing Problem (CVRP) is a classic combinatorial optimization problem with a wide range of applications in operations research. Since the CVRP is NP-hard even in a finite-dimensional Euclidean space, special attention is traditionally paid to the issues of its approximability. A major part of the known results concerning ap...

The Hitting Set Problem (HSP) is the well-known extremal problem adopting interest of researchers in the fields of statistical learning theory, combinatorial optimization, and computational geometry for decades. It is known, that the problem is NP-hard in its general case and remains intractable even in very specific geometric settings, e.g., for a...

The Cropland Capture game (CCG) aims to map cultivated lands using around 170000 satellite images. The contribution of the paper is threefold: (a) we improve the quality of the CCG’s dataset, (b) we benchmark state-of-the-art algorithms designed for an aggregation of votes in a crowdsourcing-like setting and compare the results with machine learnin...

Cropland Capture Game, a well known Geo-Wiki’s crowd-sourcing campaign, aims to map cultivated lands using around 170K satellite images of the Earth’s surface. Despite the recent progress in image analysis, the task of cropland detection is hard to automate so far since human-experts still outperform the majority of learnable machines and artificia...

The Hitting Set Problem (HSP) is the well-known extremal problem adopting interest of
researchers in the fields of statistical learning theory, combinatorial optimization, and computational
geometry for decades. It is known, that the problem is NP-hard in its general case and remains
intractable even in very specific geometric settings, e.g., for a...

The Hitting Set Problem (HSP) is the well known extremal problem adopting research interest in the fields of combinatorial optimization, computational geometry, and statistical learning theory for decades. In the general setting, the problem is NP-hard and hardly approximable. Also, the HSP remains intractable even in very specific geometric settin...

A problem of visiting megalopolises with a fixed number of “entrances” and prece- dence relations defined in a special way is studied. The problem is a natural generalization of the classical traveling salesman problem. For finding an optimal solution, we give a dynamic programming scheme, which is equivalent to a method of finding a shortest path...

The Generalized Traveling Salesman Problem (GTSP) is a combinatorial optimization problem, which is to find a minimum cost cycle visiting one point (city) from each cluster exactly. We consider a geometric case of this problem, where n nodes are given inside the integer grid (in the Euclidean plane), each grid cell is a unit square. Clusters are in...

Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem remaining NP-hard even in the Euclidean spaces of fixed dimension. Thirty years ago, in their celebrated paper, M. Haimovich and A. Rinnoy Kan proposed the first PTAS for the Planar Single Depot CVRP based on their Iterated Tour Partition heuristic. For...

We consider the combinatorial optimization problem of visiting clusters of a fixed number of nodes (cities), where, on the set of clusters should be visited according to some kind of partial order defined by additional precedence constraints. This problem is a kind of the Asymmetric Generalized Traveling Salesman Problem (AGTSP). To find an optimal...

We study the Min-k-SCCP on the partition of a complete weighted digraph by k vertex-disjoint cycles of minimum total weight. This problem is the generalization of the well-known traveling salesman problem (TSP) and the special case of the classical vehicle routing problem (VRP). It is known that the problem Min-k-SCCP is strongly NP-hard and remain...

Capacitated Vehicle Routing Problem (CVRP) is the well known intractable combinatorial optimization problem, which remains NP-hard even in the Euclidean plane. Since the introduction of this problem in the middle of the 20th century, many researchers were involved into the study of its approximability. Most of the results obtained in this field are...

The Generalized Traveling Salesman Problem (GTSP) is a combinatorial optimization problem, which is to find a minimum cost cycle visiting one point (city) from each cluster exactly. We consider a geometric case of this problem, where n nodes are given inside the integer grid (in the Euclidean plane), each grid cell is a unit square. Clusters are in...

Presentation of an invited lecture

The Cropland Capture game, which is a recently developed Geo-Wiki game, aims to map cultivated lands using around 17,000 satellite images from the Earth’s surface. Using a perceptual hash and blur detection algorithm, we improve the quality of the Cropland Capture game’s dataset. We then benchmark state-of-the-art algorithms for an aggregation of v...

The Capacitated Vehicle Routing Problem (CVRP) is a classic combinatorial optimization problem with a wide range of applications in operations research. Since the CVRP is NP-hard even in a finite-dimensional Euclidean space, special attention is traditionally paid to the issues of its approximability. A major part of the known results concerning ap...

We consider the combinatorial optimization problem of visiting clusters of a fixed number of nodes (cities), where, on the set of clusters should be visited according to some kind of partial order defined by additional precedence constraints. This problem is a kind of the Asymmetric Generalized Traveling Salesman Problem (AGTSP). To find an optimal...

A new theoretical approach to construction of efficient algo-
rithms for fingerprint image enhancement is proposed. The approach
comprises novel modifications of advanced orientation field estimation
techniques such as the method of fingerprint core extraction based on
Poincar ́e indexes and model-based smoothing for the gradient-based
approximatio...

We consider the classic setting of Capacitated Vehicle Rout-
ing Problem (CVRP): single product, single depot, demands of all customers are identical. It is known that this problem remains strongly
NP-hard even being formulated in Euclidean spaces of fixed dimension.
Although the problem is intractable, it can be approximated well in such
a special...

Crowdsourcing is a new approach to perform tasks when a group of volunteers replaces experts. For example, Geo-Wiki project [1] aims to improve the global land cover map by applying crowdsourcing for image recognition. Though crowdsourcing gives a simple way to perform tasks that are hard to automate, analysis of data received from non-experts is a...

We study the minimum-weight k-size cycle cover problem (Min-k-SCCP) of finding a partition of a complete weighted digraph into k vertex-disjoint cycles of minimum total weight. This problem is a natural generalization of the known traveling salesman problem (TSP) and has a number of applications in operations research and data analysis. We show tha...

We consider the problem of finding a fixed number of vertex-disjoint cliques of given sizes in a complete undirected weighted graph so that the total weight of vertices and edges in the cliques would be minimal. We show that the problem is strongly NP-hard both in the general case and in two subclasses, which have important applications. An approxi...

New results (along with rigorous proofs) confirming the connection between the principle of structural risk minimization and theoretical combinatorics are presented. In particular, a new special subclass of the well known Integer Partition Problem is introduced. Close relation of this subclass to pruning procedures of ensemble classifiers is proved...

For a ﬁxed natural number k a problem of k collaborating salesmen servicing the same set of cities (nodes of a given graph) is studied. We call this problem the Minimum-weight k-size cycle cover problem (or Min-k-SCCP) due to the fact that it has the following mathematical statement. Let a complete weighted digraph (with loops) be given, it is requ...

A brief survey of computational complexity and approximability results concerning efficient cluster analysis tech-niques and learning procedures in the class of piece-wise linear majority classifiers is provided. Also, new results confirming the connection between the structural minimization risk principle and theoretic combinatorics (along with ri...

A new efficient fingerprint identification algorithm combin-
ing a modification of the Delaunay triangulation minutiae-based hashing
technique for a model dataset, the Maltonian cylinder coding fingerprint
matching method, and MAP-classifier learning procedure is proposed.
Numerical experiments prove the robustness of the algorithm w.r.t. small
per...