Abstract
Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that (i) all prefix sums of the ordering are nonnegative and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. call this problem the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms. We consider a related problem, which we call the alternating stock size problem , where the numbers of positive and negative integers in the input set S are equal. The problem is the same as that shown earlier, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79. Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.
Highlights
Suppose there is a set of jobs that can be processed in any order
We show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track
Each job requires a specified amount of a particular resource, e.g. gasoline, which can be supplied in an amount chosen from a specified set of quantities
Summary
Suppose there is a set of jobs that can be processed in any order. Each job requires a specified amount of a particular resource, e.g. gasoline, which can be supplied in an amount chosen from a specified set of quantities. The goal is to order the jobs and the replenishment amounts so that the required quantity of the resource is always available for the job being processed and so that the storage space is never exceeded. (We sometimes use μ = max{μx, μy}.) Since both μx and μy are lower bounds on the value S∗ of an optimal solution, this shows that the problem can be approximated to within a factor of 2. They presented algorithms with approximation guarantees of 8/5 and 3/2 [11]
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