# Michael KhachayKrasovsky Institute of Mathematics and Mechanics Russian Academy of Sciences · Mathematical Programming Lab

Michael Khachay

Professor

Algorithms design and analysis for combinatorial optimization problems and their applications in operations research

## About

199

Publications

24,501

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

728

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)

Additional affiliations

Education

February 2005 - February 2005

September 1993 - December 1996

September 1988 - June 1993

## Publications

Publications (199)

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 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 general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there are...

Recently, O. Svensson and V. Traub have provided the first proof of the polynomial-time approximability of the asymmetric traveling salesman problem (ATSP) in the class of constant-factor approximation algorithms. Just as the famous Christofides–Serdyukov algorithm for the symmetric routing problems, these breakthrough results, applied as a “black...

Supply chain resilience is one of the most relevant topics of operations research and production management, which is aimed to risk mitigation in the global manufacturing, logistics, and trade. Conventional approach for resilient supply chain design involves the stochastic modeling and scenery-based description of anticipated failures in transporta...

In this paper, the first fixed-ratio approximation algorithms are proposed for the series of asymmetric settings of the well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all the proposed...

We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem, which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph \(G\). Each node of the graph \(G\) can either be visited by the resul...

For the first time, algorithms with constant performance guarantees are substantiated for a series of asymmetric routing problems of combinatorial optimization: the Steiner cycle problem (SCP), the generalized traveling salesman problem (GTSP), the capacitated vehicle routing problem with unsplittable customer demands (CVRP-UCD), and the prize coll...

Early knowledge about novel emerging viruses and rapid determination of their characteristics are crucial for public health. In this context, development of theoretical approaches to model viral evolution are important. The clusteron approach is a recent bioinformatics tool which analyzes genetic patterns of a specific E protein fragment and provid...

The Prize-Collecting Traveling Salesman Problem is an extension of the classic Traveling Salesman Problem, where each node of the given graph can be skipped for some known penalty. The goal is to construct a closed walk minimizing the total transportation costs and accumulated penalties. This problem has numerous applications in operations research...

The problem of optimal tool routing for CNC sheet cutting machines (referred to as Cutting Path Problem or Tool Path Problem) is considered. The general formulation is used – Generalized Segment Continuous Cutting Problem (GSCCP). The new algorithm developed by the authors to solve generalized traveling salesman problem with precedence constraints...

An instance of the Protected Shortest Simple Path Problem with Must-Pass Nodes (PSSPP-MPN) is specified by an edge-weighted directed graph with dedicated source, destination, and additional must-pass nodes. The goal is to find two vertex-disjoint paths, such that the former one is simple, visits all the must-pass nodes, and has the minimum transpor...

The Shortest Simple Path Problem with Must-Pass Nodes is the well-known combinatorial optimization problem having numerous applications in operations research. In this paper, we show, that this problem remains intractable even for any fixed number of must-pass nodes. In addition, we propose a novel problem-specific branch-and-bound algorithm for th...

The Generalized Traveling Salesman Problem (GTSP) is a well-known combinatorial optimization problem having numerous valuable practical applications in operations research. In the Precedence Constrained GTSP (PCGTSP), any feasible tour is restricted to visit all the clusters according to some given partial order. Unlike the common setting of the GT...

Viral surveillance is an essential task in public health that yields specific data, such as biological and epidemiological characteristics, crucial in the fight against viruses. We have recently developed an online platform for monitoring of Tick-Borne Encephalitis Virus, TBEV Analyzer, equipped with phylogenetic analysis and geographic map visuali...

The capacitated vehicle routing problem (CVRP) is a classical combinatorial optimization problem having a wide range of practically important applications in operations research. As most combinatorial problems, CVRP is strongly NP-hard (even on the Euclidean plane). A metric instance of CVRP is APX-complete, so it cannot be approximated to arbitrar...

The Cutting Path Problem (CPP) is a complex continuous and combinatorial optimization problem that is about finding an optimal tool path for CNC technologies equipment. The problem has many valuable industrial applications arising from the Industry 4.0 strategy, such as those, related to tool path routing for the sheet metal cutting machines. The C...

This book constitutes revised selected papers of the 9th International Conference on Analysis of Images, Social Networks and Texts, AIST 2020, held in Moscow, Russia, in october 2020. Due to the COVID-19 pandemic the conference was held online. The 14 full papers, 9 short papers and 4 poster papers were carefully reviewed and selected from 108 qual...

This book constitutes revised selected papers from the 9th International Conference on Analysis of Images, Social Networks and Texts, AIST 2020, held during October 15-16, 2020. The conference was planned to take place in Moscow, Russia, but changed to an online format due to the COVID-19 pandemic.
The 27 full papers and 4 short papers presented in...

This book constitutes the proceedings of the 20th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2021, held in Irkutsk, Russia, in July 2021.
The 29 full papers and 1 short paper presented in this volume were carefully reviewed and selected from 102 submissions. Additionally, 2 full invited papers are...

This book constitutes the refereed proceedings of the 12th International Conference on Optimization and Applications, OPTIMA 2021, held in Petrovac, Montenegro, in September-October 2021.
The 22 full and 3 short papers presented were carefully reviewed and selected from 63 submissions. The papers are organized into the following topical sub-heading...

This book constitutes the refereed proceedings of the 12th International Conference on Optimization and Applications, OPTIMA 2021, held in Petrovac, Montenegro, in September - October 2021. Due to the COVID-19 pandemic the conference was partially held online.
The 19 revised full papers presented were carefully reviewed and selected from 38 submi...

The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) is a specialized version of the well-known Generalized Traveling Salesman Problem (GTSP) having a lot of valuable applications in operations research. Despite the practical significance, results in the field of design, implementation, and numerical evaluation the algorithms...

Evaluation of the antigenic similarity degree between the strains of the influenza virus is highly important for vaccine production. The conventional method used to measure such a degree is related to performing the immunological assays of hemagglutinin inhibition. Namely, the antigenic distance between two strains is calculated on the basis of HI...

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having a host of valuable practical applications in operations research. The CVRP is strongly NP-hard both in its general case and even in very specific settings (e.g., on the Euclidean plane). The problem is APX-complete for an arbitrary metric and...

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...

In this paper, for the first time, we provide a quasi-polynomial time approximation scheme for the well-known capacitated vehicle routing problem formulated in metric spaces of an arbitrarily fixed doubling dimension

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...

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...

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...

This book constitutes the refereed proceedings of the 11th International Conference on Optimization and Applications, OPTIMA 2020, held in Moscow, Russia, in September-October 2020.*
The 21 full and 2 short papers presented were carefully reviewed and selected from 60 submissions. The papers cover such topics as mathematical programming, combinator...

This book constitutes the refereed proceedings of the 11th International Conference on Optimization and Applications, OPTIMA 2020, held in September – October 2020. Due to the COVID-19 pandemic the conference was held online.
The 18 revised full papers presented were carefully reviewed and selected from 60 submissions. The papers are organized in...

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...

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 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...

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...

We consider the Euclidean Capacitated Vehicle Routing Problem with Time Windows (CVRPTW). For the long time, approximability of this well-known problem in the class of algorithms with theoretical guarantees was poorly studied. This year, for the case of a single depot, we proposed two approximation algorithms, which are the Efficient Polynomial Tim...

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...

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 const