Anatoly Lakeyev

NP-Hard Classes of Linear Algebraic Systems with Uncertainties

For a system of linear equations Ax = b, the following natural questions appear:• does this system have a solution?• if it does, what are the possible values of a given objective function f(x1,...,xn) (e.g., of a linear function f(x) = ∑CiXi) over the systems solution set?We show that for several classes of linear equations with uncertainty (including interval linear equations) these problems are NP-hard. In particular, we show that these problems are NP-hard even if we consider only systems of n+2 equations with n variables, that have integer positive coefficients and finitely many solutions.

Linear interval equations: Computing enclosures with bounded relative or absolute overestimation is NP-hard

It is proved that for every δ>0, if there exists a polynomial-time algorithm for enclosing solutions of linear interval equations with relative (or absolute) overestimation better than δ, then P=NP. The result holds for the symmetric case as well.AbstractДоказано, что для лкхбого δ>0, еслн существуер алторнгм с-нолниомналыым временем вынолнення для локализащ ремений ннтервальной снстемы лннейных уравненщ с относнтельной (нлн абсолютной) ногрещностью, меныей δ, то P = NP. Результат снраведлнв также для случая симметричных систем.

On the computational complexity of the solution of linear systems with moduli

A problem of solvability for the system of equations of the formAx=D|x|+δ is investigated. This problem is proved to beNP-complete even in the case when the number of equations is equal to the number of variables, the matrixA is nonsingular,A≥D≥0,δ≥0, and it is initially known that the system has a finite (possibly zero) number of solutions. For an arbitrary system ofm equations ofn variables, under additional conditions that the matrixD is nonnegative and its rank is one, a polynomial-time algorithm (of the orderO((max{m, n})3)) has been found which allows to determine whether the system is solvable or not and to find one of such solutions in the case of solvability.AbstractИзучается задача разрешимости для системы уравнений видаAx=D|x|+δ. Показано, что эта задача являетсяNP-поляой даже в случае, когда число уравнений равно числу переменных, матрипаA невырождена,A≥D≥0,δ≥0 и заранее известно, что аистема имеет конечное (возможно, равное нулю) число решений. Для произвольной системыm уравнений отn переменных при донолнительном условии, что матрицаD не отрицательна и ее ранг равен единице, найден полиномиальный алгоритм (гюрядкаO((max{m, n})3)), позволяющий выяснить разрешимость этой системы и, в случае разрешимости, найти одно из решений.

Informal Introduction: Data Processing, Interval Computations, and Computational Complexity

This introduction starts with material aimed mainly at those readers who axe not well familiar with interval computations and/or with the computational complexity aspects of data processing and interval computations. It provides the motivation for the basic mathematical and computational problems that we will be analyzing in this book. Readers who are well familiar with these problems can skip the bulk of this chapter and go straight to the last section that briefly outlines the structure of the book.

On Existence and Uniqueness of Solutions of Linear Algebraic Equations in Kaucher’s Interval Arithmetic

Presently there is a growing interest to the investigation of linear algebraic equations in Kaucher’s arithmetic [1]. This interest is mainly substantiated by the fact that with the use of solutions of linear algebraic equations in Kaucher’s arithmetic allows in some cases to obtain both external and internal estimates of various sets of solutions of linear interval equations. First such results appeared rather long ago and were concerned with the external estimation of the joint set of solutions for the system of linear interval equations by algebraic solving of this system (see, e.g., G. Alefeld and J. Herzberger [2], A. Neumaier [3], and the references in these books). Later, in L.V. Kupriyanova’s [4] and S.P. Shary’s [5] works, it was shown that with the use of algebraic solutions (but already in an extended Kaucher’s interval arithmetic) it is possible to obtain both internal and external estimates for generalized sets of solutions for the systems of linear interval equations.

Solving Differential Equations

Yet another problem where interval computations axe used is a problem of solving differential equations. In this chapter, we show that in general, this problem requires at least exponential time, and briefly describe the heuristics that are used to solve important particular classes of differential equations in polynomial time.

The Notions of Feasibility and NP-Hardness: Brief Introduction

The main goal of this book is to analyze computational complexity and feasibility of data processing and interval computations. In Chapter 1, we defined the basic problems of data processing and interval computations. In this chapter, we give a brief introduction to the notions related to feasibility and computational complexity.

What if Quantities are Discrete

The basic problem of interval computations that we analyze in this book is to compute the accuracy of the results of data processing. To be more precise, we know an algorithm f(x1,...,x n) that is used in data processing and the intervals xi of possible values of the input quantities x 1,..., x n and we want to describe the set of all possible values of f(x 1,..., x n) when x i ∈ x i.

Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations

The basic problem of interval computations is to find all possible values of y = f(x1,..., xn) when we only have partial information about xi.

Properties of Interval Matrices II: Proofs and Auxiliary Results

In this chapter, we prove the results about computational complexity and feasibility of properties of interval matrices that were formulated in the previous chapter. Along the way, we also describe some important auxiliary results.