Complexity of branch-and-bound and cutting planes in mixed-integer optimization

TL;DR

This paper analyzes the theoretical computational complexity of branch-and-bound and cutting plane algorithms in mixed-integer optimization, revealing conditions where each method outperforms the other and highlighting the impact of problem structure and dimension.

Contribution

It extends complexity results to nonlinear convex problems, provides instances where cutting planes outperform branch-and-bound exponentially, and characterizes the influence of problem dimension on algorithm efficiency.

Findings

Cutting planes can exponentially outperform branch-and-bound on certain problems.

Branch-and-bound can be more efficient than cutting planes when the problem dimension is fixed.

The advantage of cutting planes diminishes outside 0/1 problems, with possible exponential gaps in complexity.

Abstract

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).