The Alternating Stock Size Problem and the Gasoline Puzzle

problem of finding a permutation of these integers such that:
(i) all prefixes of the ordering are non-negative, and
(ii) the maximum value of a prefix sum is minimized.
Kellerer et al. referred to this problem as 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 number of positive and negative integers in
the input set S are equal. The problem is the same as above, 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 Lovasz, 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.

Joint work with Heiko Röglin (Universität Bonn) and Johanna Seif (ENS Lyon).