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Given a set S of integers whose sum is zero, consider the problem of findinga permutation of these integers such that: (i) all prefix sums of the orderingare nonnegative, and (ii) the maximum value of a prefix sum is minimized.Kellerer et al. referred to this problem as the "Stock Size Problem" and showedthat it can be approximated to within 3/2. They also showed that anapproximation ratio of 2 can be achieved via several simple algorithms.  We consider a related problem, which we call the "Alternating Stock SizeProblem", where the number of positive and negative integers in the input set Sare equal. The problem is the same as above, but we are additionally requiredto alternate the positive and negative numbers in the output ordering. Thisproblem also has several simple 2-approximations. We show that it can beapproximated to within 1.79.  Then we show that this problem is closely related to an optimization versionof the gasoline puzzle due to Lov\'asz, in which we want to minimize the sizeof the gas tank necessary to go around the track. We present a 2-approximationfor this problem, using a natural linear programming relaxation whose feasiblesolutions are doubly stochastic matrices. Our novel rounding algorithm is basedon a transformation that yields another doubly stochastic matrix with specialproperties, from which we can extract a suitable permutation.

The Alternating Stock Size Problem and the Gasoline Puzzle