Research Paper#Algorithms, Computational Complexity, Graph Theory🔬 ResearchAnalyzed: Jan 3, 2026 19:12
Lower Bounds on Dynamic Programming for Connectivity Problems
Analysis
This paper provides lower bounds on the complexity of pure dynamic programming algorithms (modeled by tropical circuits) for connectivity problems like the Traveling Salesperson Problem on graphs with bounded pathwidth. The results suggest that algebraic techniques are crucial for achieving optimal performance, as pure dynamic programming approaches face significant limitations. The paper's contribution lies in establishing these limitations and providing evidence for the necessity of algebraic methods in designing efficient algorithms for these problems.
Key Takeaways
- •Establishes lower bounds on the complexity of pure dynamic programming for connectivity problems.
- •Suggests that algebraic techniques are necessary for optimal performance.
- •Uses tropical circuits to model pure dynamic programming algorithms.
- •Links tropical circuit complexity to nondeterministic communication complexity.
Reference
“Any tropical circuit calculating the optimal value of a Traveling Salesperson round tour uses at least $2^{Ω(k \log \log k)}$ gates.”