An E-approximation Scheme for Minimum Variance Problems

Abstmct Minimum variance problem may arise when we want to allocate a given amount of resource as fairly as possible to a finite set of activities under certain constraints. More formally, it is described as follows. Given a finite set E, a subset S of RE, and a function he(x(e)) from a certain domain to R for each e E E (which represents the profit resulting from ailocating x(e) amount of resource to activity e), the problem seeks to find x = {x(e) : e E E} E S that minimizes the variance of the vector {he(x(e)) : e E E}. Here the variance of {he(x(e)) : e E E} is defined as the summation over e E E of the square of difference between the profit he(x(e)) and the mean value of profits of all activities. Such problem is called minimum variance problem. This paper first presents a parametric characterization of optimal solutions. Based on this, for a class of problems satisfying certain assumptions, we shall develop an {-approximation scheme which requires to solve the corresponding parametric problem a number of times polynomial in the input length and 1/£. We shall then present three special cases for which such (-approximation scheme becomes a fully polynomial time approximation scheme. The first case is that he is linear and increasing for each e, and the feasible set S is described by the set of linear equalities and/or inequalities containing the constraint such that the sum of x(e) over all e E E is a fixed constant, and the second one is that he is linear and increasing for each e, and the feasible set S is the set of integral or real bases of sub modular systems. The third one is that he is a certain nonlinear function and the feasible set S is the set of integral or real bases of a polymatroid. Finally we shall give a pseudopolynomial time algorithm if x(e) is an integer with lower and upper bounds on it, and the sum of x(e) over all e E E is a fixed constant.