David Blokh

Maximizing traveling salesman problem for special matrices

Abstract We consider the maximizing travelling salesman problem (MTSP) for two special classes of n × n matrices with non-negative entries, namely, matrices from M(n) and M(n, α) (α ⩾ 3) defined as follows. An n × n matrix W = [wij]∈M(n) if wij = 0 for all i, j such that ¦i – j¦ ≠ 1 . An n × n matrix W = [wij]∈M(n, α) if min¦i-j¦ = 1 wij⩾αmax¦i-j¦ ≠ 1 wij. We describe an O(n)- algorithm solving exactly the MTSP for matrices fromM (n) and show that this algorithm provides an approximate solution of the MTSP for matrices from M(n, α) for α ⩾ 3 with a relative error of at most n (2α(n – 1)) . It is proved that the MTSP is NP-hard for matrices from M(n, α) for every fixed positive α.

Comparative analysis of cell parameter groups for breast cancer detection

We present a method for the comparative analysis of parameter groups according to their correlation to disease. The theoretical basis of the proposed method is Information Theory and Nonparametric Statistics. Normalized mutual information is used as the measure of correlation between parameters, and statistical conclusions are based on ranking. The fluorescence polarization (FP) parameter is considered as the principal diagnostic characteristic. The FP was measured in fluorescein diacetate (FDA)-stained individual peripheral blood mononuclear cells (PBMC), derived from healthy subjects and breast cancer (BC) patients, under different stimulation conditions: by tumor tissue, the mitogen phytohemagglutinin (PHA) or without the stimulants. The FP parameters were grouped according to their correlation with breast cancer. It was established that the greatest difference between cells of BC patients and healthy subjects is found in the PHA test (parameter P1).

The application of information theory for the research of aging and aging-related diseases

This article reviews the application of information-theoretical analysis, employing measures of entropy and mutual information, for the study of aging and aging-related diseases. The research of aging and aging-related diseases is particularly suitable for the application of information theory methods, as aging processes and related diseases are multi-parametric, with continuous parameters coexisting alongside discrete parameters, and with the relations between the parameters being as a rule non-linear. Information theory provides unique analytical capabilities for the solution of such problems, with unique advantages over common linear biostatistics. Among the age-related diseases, information theory has been used in the study of neurodegenerative diseases (particularly using EEG time series for diagnosis and prediction), cancer (particularly for establishing individual and combined cancer biomarkers), diabetes (mainly utilizing mutual information to characterize the diseased and aging states), and heart disease (mainly for the analysis of heart rate variability). Few works have employed information theory for the analysis of general aging processes and frailty, as underlying determinants and possible early preclinical diagnostic measures for aging-related diseases. Generally, the use of information-theoretical analysis permits not only establishing the (non-linear) correlations between diagnostic or therapeutic parameters of interest, but may also provide a theoretical insight into the nature of aging and related diseases by establishing the measures of variability, adaptation, regulation or homeostasis, within a system of interest. It may be hoped that the increased use of such measures in research may considerably increase diagnostic and therapeutic capabilities and the fundamental theoretical mathematical understanding of aging and disease.

An approximation algorithm with performance guarantees for the maximum traveling salesman problem on special matrices

Abstract In this paper, we investigate the maximum traveling salesman problem (Max-TSP) on quasi-banded matrices. A matrix is quasi-banded with multiplier α if all its elements contained within a band of several diagonals above and below the principal diagonal are non-zero, and any element in the band is at least α times larger than the maximal element outside the band. We investigate the properties of the Max-TSP on the quasi-banded matrices, prove that it is strongly NP-hard and derive a linear-time approximation algorithm with a guaranteed performance.

A Problem of Finding an Acceptable Variant in some Generalized Project Networks

A project network often has some activities or groups of activities which can be performed at different stages of the project. Then, the problem of finding an optimal time or/and optimal order of such an activity or a group of activities arises. Such problems emerge, in particular, in house-building management when the beginnings of some activities may vary in time or/and order. We consider a mathematical formulation of the problem, show its computational complexity and describe an algorithm for solving the problem.

The application of information theory for the estimation of old-age multimorbidity

Elderly patients are commonly characterized by the presence of several chronic aging-related diseases at once, or old-age “multimorbidity,” with critical implications for diagnosis and therapy. However, at the present there is no agreed or formal method to diagnose or even define “multimorbidity.” There is also no formal quantitative method to evaluate the effects of individual or combined diagnostic parameters and therapeutic interventions on multimorbidity. The present work outlines a methodology to provide such a measurement and definition, using information theoretical measure of normalized mutual information. A cohort of geriatric patients, suffering from several age-related diseases (multimorbidity), including ischemic heart disease, COPD, and dementia, were evaluated by a variety of diagnostic parameters, including static as well as dynamic biochemical, functional-behavioral, immunological, and hematological parameters. Multimorbidity was formally coded and measured as a composite of several chronic age-related diseases. The normalized mutual information allowed establishing the exact informative value of particular parameters and their combinations about the multimorbidity value. With the currently intensifying attempts to reduce aging-related multimorbidity by therapeutic interventions into its underlying aging processes, the proposed method may outline a valuable direction toward the formal indication and evidence-based evaluation of effectiveness of such interventions.