Greedy can beat pure dynamic programming

TL;DR

This paper demonstrates that greedy algorithms can outperform pure dynamic programming in certain optimization problems, establishing an exponential lower bound on pure DP for minimum spanning trees and highlighting their computational differences.

Contribution

It proves an exponential lower bound for pure DP algorithms solving minimum spanning trees, showing their limitations compared to greedy methods.

Findings

Pure DP algorithms require exponential operations for MSTs.

Greedy algorithms like Kruskal are efficient and can outperform pure DP.

Pure DP and greedy algorithms have incomparable computational powers.

Abstract

Many dynamic programming algorithms for discrete 0-1 optimizationproblems are "pure" in that their recursion equations only use min/max and addition operations, and do not depend on actual input weights. 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 for this problem must perform $2^{\Omega(\sqrt{n})}$ operations. Since the greedy algorithm can also badly fail on some optimization problems, easily solvable by pure DP algorithms, our result shows that the computational powers of these two types of algorithms are incomparable.