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## Polynomial-time approximation algorithms for the coloring problem in some cases

Journal of Combinatorial Optimization. 2017. Vol. 33. No. 3. P. 809-813.

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.

Language:
English

Malyshev D., Pardalos P. M., Optimization Letters 2016 Vol. 10 No. 8 P. 1593-1612

The task of complete complexity dichotomy is to clearly distinguish between easy and hard cases of a given problem on a family of subproblems. We consider this task for some optimization problems restricted to certain classes of graphs closed under deletion of vertices. A concept in the solution process is based on revealing the so-called ...

Added: December 18, 2015

Malyshev D., Discrete Applied Mathematics 2018 Vol. 247 P. 423-432

We show that the weighted coloring problem can be solved for {P5,banner}-free graphs and for {P5,dart}-free graphs in polynomial time on the sum of vertex weights. ...

Added: April 23, 2018

Malyshev D., Razvenskaya O., Discrete Applied Mathematics 2017 Vol. 219 P. 158-166

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. ...

Added: November 21, 2016

Malyshev D., Pardalos P. M., Doklady Mathematics 2014 Vol. 89 No. 2 P. 253-256

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an
optimal solution and it is a base for design of branch-and-bound algorithms solving such problems. We show in this paper that for the weighted independent set problem and a bipartite graph with n vertices and ...

Added: April 18, 2014

Malyshev D., Pardalos P. M., Optimization Letters 2015 Vol. 9 No. 5 P. 839-843

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 ...

Added: September 26, 2014

Malyshev D., Journal of Combinatorial Optimization 2016 Vol. 32 No. 1 P. 226-243

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with ...

Added: April 4, 2015

Goldengorin B. I., Malyshev D., Pardalos P. M., Doklady Mathematics 2013 Vol. 87 No. 3 P. 368-371

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design branch-and-bound algorithms solving such problems. In this paper we show that for the weighted independent set problem on trees with n vertices all upper and ...

Added: June 23, 2013

Kontchakov R., Pratt-Hartmann I., Nenov Y. et al., ACM Transactions on Computational Logic 2013 Vol. 14 No. 2 P. 13.1-13.48

We consider the quantifier-free languages, Bc and Bc°, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of Rn (n ≥ 2) and, additionally, over the regular closed ...

Added: March 25, 2015

Malyshev D., Discrete Mathematics 2015 Vol. 338 No. 11 P. 1860-1865

We completely determine the complexity status of the 3-colorability problem for hereditary graph classes defined by two forbidden induced subgraphs with at most five vertices. ...

Added: April 7, 2014

Gafarov E., Dolgui A., Lazarev A. A., / Elsevier. Series -- "Computers & Industrial Engineering". 2014.

In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains ...

Added: April 10, 2015

Malyshev D., / Cornell University. Series math "arxiv.org". 2013. No. 1307.0278v1.

The coloring problem is studied in the paper for graph classes deﬁned by two small forbidden induced subgraphs. We prove some suﬃcient conditions for eﬀective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most ﬁve vertices except ...

Added: October 3, 2013

Malyshev D., Optimization Letters 2021 Vol. 15 No. 2 P. 311-326

The vertex colourability problem is to determine, for a given graph and a given natural k, whether it is possible to split the graph’s vertex set into at most k subsets, each of pairwise non-adjacent vertices, or not. A hereditary class is a set of simple graphs, closed under deletion of vertices. Any such a class can be ...

Added: January 6, 2021

Malyshev D., Journal of Combinatorial Optimization 2014 Vol. 27 No. 2 P. 345-354

The notion of a boundary graph class was recently introduced for a classification of hereditary graph classes according to the complexity of a considered problem. Two concrete graph classes are known to be boundary for several graph problems. We formulate a criterion to determine whether these classes are boundary for a given graph problem or ...

Added: February 7, 2013

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2013 Vol. 7 No. 2 P. 221-228

The notion of a boundary class of graphs is a helpful tool for the computational complexity analysis of graph theory problems in the family of hereditary classes. Some general and specific features for families of boundary classes of graphs for the vertex k-colorability problem and its “limit” variant, the chromatic index problem, were studied by ...

Added: June 23, 2013

Sirotkin D., Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2018 Vol. 12 No. 4 P. 759-769

The 3-coloring problem for a given graph consists in verifying whether it is possible
to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete
complexity classification is known for this problem for the hereditary classes defined by triples of
forbidden induced subgraphs, each on at most 5 vertices. In this article, ...

Added: November 20, 2018

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2012 Vol. 6 No. 1 P. 97-99

Under study is the complexity status of the independent set problem in a class of connected graphs that are defined by functional constraints on the number of edges depending on the number of vertices. For every natural number C, this problem is shown to be polynomially solvable in the class of graphs, On the other ...

Added: December 7, 2012

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2013 Vol. 7 No. 3 P. 412-419

The notion is introduced of an expanding operator for the independent set problem. This notion is a useful tool for the constructive formation of new cases with the efficient solvability of the problem in the family of hereditary classes of graphs and is applied to hereditary parts of the set Free({P_5,C_5}). It is proved that ...

Added: October 3, 2013

Artale A., Kontchakov R., Ryzhikov V. et al., ACM Transactions on Computational Logic 2014 Vol. 15 No. 3 P. 25.1-25.50

We design temporal description logics (TDLs) suitable for reasoning about temporal conceptual data models and investigate their computational complexity. Our formalisms are based on DL-Lite logics with three types of concept inclusions (ranging from atomic concept inclusions and disjointness to the full Booleans), as well as cardinality constraints and role inclusions. The logics are interpreted ...

Added: March 25, 2015

Malyshev D., Дискретный анализ и исследование операций 2012 Т. 19 № 3 С. 58-64

An algorithm is implemented in the article for finding the independence number of a n-vertex graph from the class Free({P5,C5, Kp}) in time O(np+O(1)). ...

Added: June 6, 2012

Shvydun S., / Высшая школа экономики. Series WP7 "Математические методы анализа решений в экономике, бизнесе и политике". 2015. No. WP7/2015/07.

Two-stage superposition choice procedures, which sequentially apply two choice procedures so that the result of the first choice procedure is the input for the second choice procedure, are studied. We define which of them satisfy given normative conditions, showing how a final choice is changed due to the changes of preferences or a set of ...

Added: October 20, 2015

Kazda A., Opršal J., Valeriote M. et al., Canadian Mathematical Bulletin 2020 P. 577-591

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation m that satisfies the minority equations m(y,x,x)≈m(x,y,x)≈m(x,x,y)≈y . We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP. ...

Added: June 15, 2020

Malyshev D., Siberian Electronic Mathematical Reports 2014 Vol. 11 P. 811-822

We obtain a complete complexity dichotomy for the edge 3- colorability within the family of hereditary classes defined by forbidden
induced subgraphs on at most 6 vertices and having at most two 6-vertex forbidden induced structures. ...

Added: April 7, 2014

Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2020 Vol. 14 No. 4 P. 706-721

The edge coloring problem for a graph is to minimize the number of colors that are sufficient to color all edges of the graph so that all adjacent edges receive distinct colors. The computational complexity of the problem is known for all graph classes defined by forbidden subgraphs with at most 6 edges. We improve ...

Added: January 30, 2021

Gribanov D., Malyshev D., Mokeev D. B., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2020 Vol. 14 P. 480-489

We consider the problem of minimizing the number of colors in the colorings of the vertices of a given graph so that, to each vertex there is assigned some set of colors whose number is equal to the given weight of the vertex; and adjacent vertices receive disjoint sets. For all hereditary classes defined by ...

Added: October 30, 2020