Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties -- those that cannot be characterized by a finite set of forbidden subgraphs -- have quantum query complexity Theta(n^(3/2)). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^(3/2)) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.
quatum query complexity
quantum algorithms
lower bounds
graph minors
graph properties
661-672
Regular Paper
Andrew M.
Childs
Andrew M. Childs
Robin
Kothari
Robin Kothari
10.4230/LIPIcs.STACS.2011.661
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode