Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs

TL;DR

This paper extends lower bounds for local search query complexity from specific graphs to all vertex-transitive graphs, providing new bounds for randomized and quantum algorithms based on graph size and diameter.

Contribution

It generalizes Aaronson's techniques to establish lower bounds for local search on all vertex-transitive graphs, not just hypercubes and grids.

Findings

Randomized query complexity is at least Ω(N^{1/2}/d log N).

Quantum query complexity is at least Ω(N^{1/4}/√(d log N)).

Bounds depend on graph size N and diameter d.

Abstract

We study the problem of \emph{local search} on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires $\Omega(2^{n/2 - o(1)})$ queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to $\Omega(2^{n/2}/n^2)$ and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are $\Omega(N^{1/2}/d\log N)$ and $\Omega(N^{1/4}/\sqrt{d\log N})$, respectively.