Artem V. Pyatkin

Graphs capturing alternations in words

A graph G = (V,E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x, y) ∈ E for each x ≠ y. If W is k-uniform (each letter of W occurs exactly k times in it) then G is called k-representable. A graph is representable if and only if it is k-representable for some k [1].

Semi-transitive orientations and word-representable graphs

A graph G = ( V , E ) is a word-representable graph if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if ( x , y ) ? E for each x ? y .In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a semi-transitive orientation defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs.We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2 n , while there exist graphs for which it is n / 2 .

A 2-approximation algorithm for the metric 2-peripatetic salesman problem

In the m-peripatetic traveling salesman problem (m-PSP), given an n-vertex complete undirected edge-weighted graph, it is required to find m edge disjoint Hamiltonian cycles of minimum total weight. The problem was introduced by Krarup (1974) and has network design and scheduling applications. It is known that 2-PSP is NP-hard even in the metric case and does not admit any constant-factor approximation in the general case. Baburin, Gimadi, and Korkishko (2004) designed a (9/4 + Ɛ)-approximation algorithm for the metric case of 2-PSP, based on solving the traveling salesman problem. In this paper we present an improved 2-approximation algorithm with running time O(n2 log n) for the metric 2-PSP. Our algorithm exploits the fact that the problem of finding two edge disjoint spanning trees of minimum total weight is polynomially solvable.

Approximation polynomial algorithm for the data editing and data cleaning problem

The work considers the mathematical aspects of one of the most fundamental problems of data analysis: search (choice) among a collection of objects for a subset of similar ones. In particular, the problem appears in connection with data editing and cleaning (removal of irrelevant (not similar) elements). We consider the model of this problem, i.e., the problem of searching for a subset of maximal cardinality in a finite set of points of the Euclidean space for which quadratic variation of points with respect to its unknown centroid does not exceed a given fraction of the quadratic variation of points of the input set with respect to its centroid. It is proved that the problem is strongly NP-hard. A polynomial 1/2-approximation algorithm is proposed. The results of the numerical simulation demonstrating the effectiveness of the algorithm are presented.

Completing Partial Schedules for Open Shop with Unit Processing Times and Routing

Open Shop is a classical scheduling problem: given a seti¾ź

Complexity of the weighted max-cut in Euclidean space

mathcal J

Word-Representable Graphs: a Survey

of jobs and a seti¾ź