Quantum Lower Bound for Graph Collision Implies Lower Bound for Triangle Detection

TL;DR

This paper establishes a connection between quantum lower bounds for the GRAPH-COLLISION problem and the TRIANGLE detection problem, suggesting that improvements in one could lead to better bounds in the other.

Contribution

It demonstrates that an improved quantum lower bound for GRAPH-COLLISION would imply a better lower bound for TRIANGLE detection, linking the complexity of these two problems.

Findings

Current lower bounds are trivial for both problems

Improvement in one problem's lower bound affects the other

No known matching upper bounds for these problems

Abstract

We show that an improvement to the best known quantum lower bound for GRAPH-COLLISION problem implies an improvement to the best known lower bound for TRIANGLE problem in the quantum query complexity model. In GRAPH-COLLISION we are given free access to a graph $(V,E)$ and access to a function $f:V\rightarrow \{0,1\}$ as a black box. We are asked to determine if there exist $(u,v) \in E$, such that $f(u)=f(v)=1$. In TRIANGLE we have a black box access to an adjacency matrix of a graph and we have to determine if the graph contains a triangle. For both of these problems the known lower bounds are trivial ($\Omega(\sqrt{n})$ and $\Omega(n)$, respectively) and there is no known matching upper bound.