The Alternating Stock Size Problem and the Gasoline Puzzle
Article No.: 19, Pages 1 - 23
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.
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Index Terms
- The Alternating Stock Size Problem and the Gasoline Puzzle
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April 2018
339 pages
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Association for Computing Machinery
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Publication History
Published: 16 April 2018
Accepted: 01 December 2017
Revised: 01 November 2017
Received: 01 December 2016
Published in TALG Volume 14, Issue 2
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- European Research Council under ERC
- LabEx PERSYVAL-Lab
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