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

TL;DR

This paper establishes strong lower bounds on the complexity of dynamic programming algorithms for connectivity problems like TSP on graphs with bounded pathwidth, showing algebraic methods are inherently costly.

Contribution

It provides unconditional parameterized complexity lower bounds for tropical circuit models of dynamic programming on graphs with small pathwidth, surpassing existing algorithm efficiencies.

Findings

Any tropical circuit solving TSP on certain bounded pathwidth graphs requires exponential size.

Lower bounds are linked to nondeterministic communication complexity of compatibility matrices.

Results suggest algebraic dynamic programming approaches are fundamentally limited in worst-case scenarios.

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 $2^{\Omega(k \log \log k)}$ 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.