## 05.03.2024 16:00 Maximilian Gläser: Sub-Exponential Lower Bounds for Branch-and-Bound with General Disjunctions via Interpolation

We investigate linear programming based branch-and-bound using general disjunctions, also known as stabbing planes and derive the first sub-exponential lower bound (i.e., 2^{L^\omega(1)}) in the encoding length L of the integer program for the size of a general branch-and-bound tree. In fact, this is even the first super-linear bound.
This is achieved by showing that general branch-and-bound admits quasi-feasible monotone real interpolation. Specialized to a certain infeasible integer program whose infeasibility expresses that no graph can have both a k-coloring and a (k+1)-clique, this property states that any branch-and-bound tree for this program can be turned into a monotone real circuit of similar size which distinguishes between graphs with a k-coloring and graphs with a (k+1)-clique. Thus, we can utilize known lower-bounds for such circuits.
Using Hrubeš' and Pudlák's notion of infeasibility certificates, one can show that certain random CNFs are sub-exponentially hard to refute for branch-and-bound with high probability via the same technique.
This is joint work with Marc Pfetsch.

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