On a labeling problem in graphs

AbstractMotivated by applications in software programming, we consider the problem of covering a graph by a feasible labeling. Given an undirected graph G=(V,E), two positive integers k and t, and an alphabet Σ, a feasible labeling is defined as an assignment of a set Lv⊆Σ to each vertex v∈V, such that (i) |Lv|≤k for all v∈V and (ii) each label α∈Σ is used no more than t times. An edge e={i,j} is said to be covered by a feasible labeling if Li∩Lj≠0̸. G is said to be covered if there exists a feasible labeling that covers each edge e∈E.In general, we show that the problem of deciding whether or not a tree can be covered is strongly NP-complete. For k=2, t=3, we characterize the trees that can be covered and provide a linear time algorithm for solving the decision problem. For fixed t, we present a strongly polynomial algorithm that solves the decision problem; if a tree can be covered, then a corresponding feasible labeling can be obtained in time polynomial in k and the size of the tree. For general graphs, we give a strongly polynomial algorithm to resolve the covering problem for k=2, t=3

LRS Bianchi Type V Universe in Creation-Field Cosmology

Abstract. The Hoyle-Narlikar C-field cosmology with LRS Bianchi type-V space-time has been studied. Using methods of Narlikar and Padmanabham [1], the solutions have been obtained when the creation field C is a function of time t only. The geometrical and physical aspects of the model are also examined

Изучение структуры цинктитановых боросиликатных стекол по данным рассеяния рентгеновских лучей под малыми углами

В статті вивчено мікронеоднорідну будову цинктитанових боросилікатних стекол та процесів фазового розділу в них за даним розсіяння нейтронів під малими кутами. Зроблено висновок про характер розподілення часток, що виділяються за розмірами, які змінюються в дослідних стеклах в залежності від вмісту в них TiO₂ та ZnO. Встановлено вплив наявності мікронеоднорідностей після варки на характер їх фазовогорозподілення.In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a leading-out on distribution of depositing corpuscles character on sizes which changes in studied glasses depending on the contents in them TiO₂ and ZnO. Effect of presence micro micronon-uniforms after melting on character of their phase separationis established

Approximation Algorithms for Scheduling with Reservations

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan Cmax, which is the maximum completion time. The general case of the problem is inapproximable unless P = NP; hence we study a suitable strongly NP-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemeted by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless P = NP. The PTAS presented here is the rst one for the problem under consideration; so far, not even for well-know special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a vonstant approximation ratio and discuss FPTAes for special cases as well as the complexity of the problem if m is part of the input

Structural analysis of a fractional matching problem

AbstractMixed Software Programming refers to a novel software development paradigm resulting from efforts to combine two different programming approaches: Solo Programming and Pair Programming. Solo Programming refers to the traditional practice of assigning a single developer to develop a software module and Pair Programming refers to a relatively new approach where two developers work simultaneously on developing a module. In Mixed Programming, given a set of modules to be developed, a chosen subset of modules may be developed using Solo Programming and the remaining modules using Pair Programming.Motivated by applications in Mixed Software Programming, we consider the following generalization of classical fractional 1-matching problem: Given an undirected simple graph G=(V;E), and a positive number F, find values for xe,e∈E, satisfying the following: 1.x∈{0,12,1}∀e∈E.2.∑e∈δ(i)xe≤1∀i∈V, where δ(i)={e∈E:e=(i,j)},i∈V.3.Maximize {2∑e∈Exe−F|{i∈V:∑e∈δ(i)xe=1}|}. We show that this problem is solvable in strongly polynomial time. Our primary focus in this paper is on obtaining the structure of the optimal solution for an arbitrary instance of the problem

On the structure of bottlenecks in processes

202307 bcchAccepted ManuscriptSelf-fundedPublishe