  Let G=(V,E) be a finite graph, and f:V->N be any function. The Local Search
problem consists in finding a local minimum of the function f on G, that is a
vertex v such that f(v) is not larger than the value of f on the neighbors of v
in G. In this note, we first prove a separation theorem slightly stronger than
the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This
result allows us to enhance a previously known deterministic algorithm for
Local Search with query complexity O(\log n)\cdot d+O(\sqrt{g})\cdot\sqrt{n},
so that we obtain a deterministic query complexity of
d+O(\sqrt{g})\cdot\sqrt{n}, where n is the size of G, d is its maximum degree,
and $g$ is its genus. We also give a quantum version of our algorithm, whose
query complexity is of O(\sqrt{d})+O(\sqrt[4]{g})\cdot\sqrt[4]{n}\log\log n.
Our deterministic and quantum algorithms have query complexities respectively
smaller than the generic algorithms of Aldous and of Aaronson for large classes
of graphs, including graphs of bounded genus and planar graphs. Independently
from this work, Zhang has recently given a quantum algorithm which finds a
local minimum on the planar grid over \{1,...,\sqrt{n}\}^2 using
O(\sqrt[4]{n}(\log\log n)^2) queries. Our quantum algorithm can be viewed as a
strongly generalized, and slightly enhanced version of this algorithm.
