The problem of finding a local minimum of a black-box function is central
for understanding local search as well as quantum adiabatic algorithms.
For functions on the Boolean hypercube {0,1}^n, we show a lower bound of
Omega(2^{n/4}/n) on the number of queries needed by a quantum computer to
solve this ... more >>>

The Local Search problem, which finds a
local minimum of a black-box function on a given graph, is of both
practical and theoretical importance to many areas in computer
science and natural sciences. In this paper, we show that for the
Boolean hypercube $\B^n$, the randomized query complexity of Local
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2-Opt is probably the most basic and widely used local search
heuristic for the TSP. This heuristic achieves amazingly good
results on "real world" Euclidean instances both with respect to
running time and approximation ratio. There are numerous
experimental studies on the performance of 2-Opt. However, the
theoretical knowledge about ... more >>>