Approximation Limitations of Pure Dynamic Programming

TL;DR

This paper establishes super-polynomial lower bounds on the size of tropical circuits, showing fundamental limitations of pure dynamic programming algorithms in approximating optimization problems.

Contribution

It provides the first super-polynomial lower bounds for tropical circuits, highlighting the inherent approximation limitations of pure DP algorithms.

Findings

Super-polynomial lower bounds for tropical circuits

Pure DP algorithms and greedy algorithms have incomparable approximation powers

Limitations of pure DP algorithms in approximation tasks

Abstract

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.