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## The Complexity of Positive Semidefinite Matrix Factorization

SIAM Journal on Optimization. 2017. Vol. 27. No. 3. P. 1898-1909.

Shitov Y.

Let $A$ be an $m\times n$ matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B_1,\ldots,B_m$, $C_1,\ldots,C_n$ satisfying $A(i|j)=\operatorname{tr}(B_iC_j)$ for all $i,j$. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.

Shitov Y., SIAM Journal on Optimization 2018 Vol. 28 No. 3 P. 2067-2072

Gouveia, Parrilo, and Thomas gave a description of certain rank functions of matrices in geometric terms, generalizing a celebrated result of Yannakakis on the nonnegative rank. We analyze the algorithmic complexity of their description using the results of Renegar on the first-order theory of the reals. This gives a proof that matrices whose positive semidefinite ...

Added: September 26, 2018

Shitov Y., St Petersburg Mathematical Journal 2015 Vol. 26 P. 341-350

In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules S in R^n having topological dimension two are studied, and it is shown that any such S has a finite weak dimension not exceeding n. For a fixed k, a polynomial time algorithm is constructed that ...

Added: August 12, 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

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

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., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2017 Vol. 11 No. 1 P. 99-106

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 ...

Added: February 13, 2017

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., Дискретный анализ и исследование операций 2012 Т. 19 № 6 С. 37-48

Понятие граничного класса графов является полезным инструментом для анализа вычислительной сложности задач на графах в семействе наследственных классов. В предыдущих работах автора исследовались общие черты и особенности семейств граничных классов графов для задачи о вершинной k-раскраске и ее «предельного варианта» - задачи о хроматическом числе. В данной работе эта проблематика рассматривается применительно к реберному варианту ...

Added: November 30, 2012

Korpelainen N., Lozin V. V., Malyshev D. et al., Theoretical Computer Science 2011 No. 412 P. 3545-3554

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle ...

Added: September 11, 2012

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

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

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

Shitov Y., Theoretical Computer Science 2017 Vol. 660 P. 102-104

Let G be a simple graph whose vertices are partitioned into two subsets, called ‘filled’ vertices and ‘empty’ vertices. A vertex v is said to be forced by a filled vertex u if v is a unique empty neighbor of u. If we can fill all the vertices of G by repeatedly filling the forced ones, then we call an initial set of filled vertices a ...

Added: December 19, 2016

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

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., 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

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

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

Rybakov M., Shkatov D., Journal of Logic and Computation 2021 Vol. 31 No. 2 P. 426-443

It is shown that products and expanding relativized products of propositional modal logics where one component is the minimal monomodal logic K are polynomial-time reducible to their single-variable fragments. Therefore, the nown lower bound complexity and undecidability results for such logics are extended to their single-variable fragments. Similar results are obtained for products where one component is a polymodal logic with a K-style ...

Added: September 24, 2020

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

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., 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

Berlin : Springer, 2012

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 ...

Added: October 30, 2013