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.

翻译：我们证明了第一种,甚至是超极球性,在热带(min,+)和(max,+)电路大小的下限线上,相对称得上优化问题。许多典型的优化问题动态编程(DP)算法是纯纯的,因为它们只使用基本分钟,最大,+ 运行在循环方程式中。热带电路是这类算法的严格数学模型。我们热带电路下限的算法后果是纯DP算法和贪婪算法的近似能力是无法比较的。纯DP算法在近似能力上几乎无法战胜近似上的贪婪,这一点早已为人所熟知。新的结果是,还存在相反的立点。