A Two-Resource Allocation Problem Solvable in Linear Time

Introduction. The complexity of the linear programming problem is a central question in the interface between operations research and computer science. It is still an open question whether an m x n problem can be solved within a polynomial number p(m, n) of arithmetic operations. Another direction of research is that of exploring the complexities of structured linear programming problems (see [M 1, M2, M31). It turns out that for many subproblems there exist algorithms which are more efficient than the simplex algorithm and investigating such algorithms may eventually lead to an alternative for the simplex algorithm in the general case. In this paper we present a linear-time algorithm for a class of linear programming problems which may be described as follows. Suppose we have n independent "tasks" denoted 1,2, . . . , n, whose processing times depend on the amounts of two resources allocated to them. More specifically, assume we have X units of the first resource and Y units of the second resource. Given the initial time t,, if we allocate x, units of the first resource and y, units of the second resource to task i, then task i will process in T, = max{ t, a,x, b,y,, d l ) time units ( t , , a,, b,, dl > 0). Suppose we wish to minimize makespan [B], given all tasks start at time zero. Then our problem is to minimize Max{ T , ) subject to E x , = X , Cy, = Y and x,, y, > 0. It is easy to see that it is sufficient to minimize Max{ t, a,x, b,y,). For some references on resource allocation consult [El, [GM] and [KIMI], [KIM2]. Recent results of Megiddo [Ml],[M2] establish that whenever the number of variables (or, equivalently, the number of constraints) is fixed, a linear programming problem is solvable in linear time. However, the problem we discuss in this paper does not fall in this category. The methods we use are related to those used in [Ml, M2]. Our problem is also related but not equivalent to the multiple-choice linear knapsack problem [IHTI], [Zl] recently revisited by Dyer [D3] and Zemel [Z2], the latter two stemming from the results independently obtained by Megiddo [MI], [M2] and Dyer [Dl],[D2]. Thus, it turns out. that these new methods lead to better algorithms than previous ones. Apparently, the "sorting" step can be avoided in these special linear programming problems and successive reductions in the size of the problem yield