  We investigate the theoretical complexity of branch-and-bound (BB) and
cutting plane (CP) algorithms for mixed-integer optimization. In particular, we
study the relative efficiency of BB and CP, when both are based on the same
family of disjunctions. We extend a result of Dash to the nonlinear setting
which shows that for convex 0/1 problems, CP does at least as well as BB, with
variable disjunctions. We sharpen this by giving instances of the stable set
problem where we can provably establish that CP does exponentially better than
BB. We further show that if one moves away from 0/1 sets, this advantage of CP
over BB disappears; there are examples where BB finishes in O(1) time, but CP
takes infinitely long to prove optimality, and exponentially long to get to
arbitrarily close to the optimal value (for variable disjunctions). We next
show that if the dimension is considered a fixed constant, then the situation
reverses and BB does at least as well as CP (up to a polynomial blow up), no
matter which family of disjunctions is used. This is also complemented by
examples where this gap is exponential (in the size of the input data).
