纯动态规划的逼近极限
Approximation Limitations of Pure Dynamic Programming
摘要：
我们证明了第一，甚至超多项式，下界的大小热带（min，+）和（max，+）电路近似给定的优化问题。许多经典的动态规划（DP）优化算法都是纯算法，它们在递归方程中只使用基本的min、max、+运算。热带电路构成了这类算法的严格数学模型。我们的热带电路的下界的算法的结果是，纯DP算法和贪心算法的近似能力是不可比拟的。众所周知，纯DP算法在近似方面很难击败贪婪算法。这个结论的新之处在于，反过来也成立。
英文摘要：
We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+) and (max,+) circuits approximating given optimization problems. Many classical dynamic programming (DP) algorithms for optimization problems are pure in that they only use the basic min, max, + operations in their recursion equations. Tropical circuits constitute a rigorous mathematical model for this class of algorithms. An algorithmic consequence of our lower bounds for tropical circuits is that the approximation powers of pure DP algorithms and greedy algorithms are incomparable. That pure DP algorithms can hardly beat greedy in approximation, is long known. New in this consequence is that also the converse holds.
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