Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a set S of integers whose sum is zero, consider the 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 shod that it can be approximated to within 3/2. They also shod that an approximation ratio of 2 can be achieved via several simple algorithms.#R##N##R##N# consider a related problem, which  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  are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations.   show that it can be approximated to within 1.79.#R##N##R##N#Then  show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovasz, in which  want to minimize the size of the gas tank necessary to go around the track.  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  can extract a suitable permutation.
['Alantha Newman', 'Heiko Röglin', 'Johanna Seif']
I am writing a research paper on a idea The Alternating Stock Size Problem and the Gasoline Puzzle
[{"from": "human", "value": "I am writing a research paper on a idea The Alternating Stock Size Problem and the Gasoline Puzzle"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a set S of integers whose sum is zero, consider the 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 shod that it can be approximated to within 3/2. They also shod that an approximation ratio of 2 can be achieved via several simple algorithms.#R##N##R##N# consider a related problem, which  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  are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations.   show that it can be approximated to within 1.79.#R##N##R##N#Then  show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovasz, in which  want to minimize the size of the gas tank necessary to go around the track.  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  can extract a suitable permutation."}]