Quantum and randomized lower bounds for local search on vertex-transitive graphs

We study the problem of local search on a graph. Given a real-valued black-box functionf 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. In1983, Aldous gave the first strong lower bounds for the problem, showing that anyrandomized algorithm requires Ω(2n/2-o(n)) queries to determine a local minimum onthe n-dimensional hypercube. The next major step forward was not until 2004 whenAaronson, introducing a new method for query complexity bounds, both strengthened thislower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum querycomplexity. While these bounds are very strong, they are known only for narrow familiesof graphs (hypercubes and grids). We show how to generalize Aaronson's techniques inorder to give randomized (and quantum) lower bounds on the query complexity of localsearch for the family of vertex-transitive graphs. In particular, we show that for anyvertex-transitive graph G of N vertices and diameter d, the randomized and quantumquery complexities for local search on G are Ω (√N/dlogN) and (4√N / √dlogN),respectively.