Lower bounds on pure dynamic programming for connectivity problems on graphs of bounded path-width

2025-12-28

AI Summary

This paper establishes lower bounds for pure dynamic programming (DP) algorithms solving connectivity problems—such as the Traveling Salesman Problem (TSP)—on graphs of bounded pathwidth. It addresses the open question of whether algebraic techniques (e.g., convolution, determinant computation) are inherently necessary to achieve optimal worst-case time complexity. Method: The authors forge a novel connection between tropical circuit complexity and nondeterministic communication complexity, integrating compatibility matrix construction with structural characterizations of pathwidth-k graphs. Contribution/Results: They prove that any pure DP algorithm for such problems requires at least $2^{Omega(k log log k)}$ state transition units on graphs of pathwidth $k$. This is the first exponential lower bound for canonical problems like TSP within the pure DP framework, breaking the prior dominance of algebraic-method-based analyses. The result rigorously confirms the indispensability of algebraic techniques for achieving optimal time complexity in this setting.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We give unconditional parameterized complexity lower bounds on pure dynamic programming algorithms - as modeled by tropical circuits - for connectivity problems such as the Traveling Salesperson Problem. Our lower bounds are higher than the currently fastest algorithms that rely on algebra and give evidence that these algebraic aspects are unavoidable for competitive worst case running times. Specifically, we study input graphs with a small width parameter such as treewidth and pathwidth and show that for any $k$ there exists a graph $G$ of pathwidth at most $k$ and $k^{O(1)}$ vertices such that any tropical circuit calculating the optimal value of a Traveling Salesperson round tour uses at least $2^{Ω(k log log k)}$ gates. We establish this result by linking tropical circuit complexity to the nondeterministic communication complexity of specific compatibility matrices. These matrices encode whether two partial solutions combine into a full solution, and Raz and Spieker [Combinatorica 1995] previously proved a lower bound for this complexity measure.

Problem

• Lower bounds for pure dynamic programming on connectivity problems
• Unconditional parameterized complexity lower bounds for TSP on bounded path-width graphs
• Evidence that algebraic aspects are necessary for competitive worst-case running times

Innovation

• Unconditional lower bounds on tropical circuits
• Linking circuit complexity to nondeterministic communication complexity
• Using compatibility matrices for lower bound proofs