eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2011-03-11
661
672
10.4230/LIPIcs.STACS.2011.661
article
Quantum query complexity of minor-closed graph properties
Childs, Andrew M.
Kothari, Robin
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol009-stacs2011/LIPIcs.STACS.2011.661/LIPIcs.STACS.2011.661.pdf
quatum query complexity
quantum algorithms
lower bounds
graph minors
graph properties