It is known that the size of monotone arithmetic (+,\ast) circuits can be exponentially decreased by allowing just one division "at the very end," at the output gate. A natural question is: can the size of (+,\ast) circuits be substantially reduced if we allow divisions "at the very beginning," that ... more >>>
We develop general lower bound arguments for approximating tropical
(min,+) and (max,+) circuits, and use them to prove the
first non-trivial, even super-polynomial, lower bounds on the size
of such circuits approximating some explicit optimization
problems. In particular, these bounds show that the approximation
powers of pure dynamic programming algorithms ...
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Many dynamic programming algorithms are ``pure'' in that they only use min or max and addition operations in their recursion equations. The well known greedy algorithm of Kruskal solves the minimum weight spanning tree problem on n-vertex graphs using only O(n^2\log n) operations. We prove that any pure DP algorithm ... more >>>