  We apply classical algorithms for approximately solving constraint
satisfaction problems to find bounds on extremal eigenvalues of local
Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on
the number of terms in which each spin participates, and find extensive bounds
for the operator norm and ground-state energy of such Hamiltonians under this
constraint. In each case the bound is achieved by a product state which can be
found efficiently using a classical algorithm.
