branch and bound verification

**Branch and Bound Verification** is the **core algorithmic paradigm for exact neural network verification** — systematically partitioning the input space (branching) and computing bounds on each subregion (bounding) to either prove or disprove a property. **How Branch and Bound Works** - **Bounding**: Use relaxation methods (LP, IBP, CROWN) to compute output bounds for a given input region. - **Decision**: If bounds prove the property → verified. If bounds show a violation → counterexample found. - **Branching**: If bounds are inconclusive, split the input region (or split a ReLU activation state) into sub-problems. - **Pruning**: Sub-problems that are provably safe (from bounding) are pruned — no further branching needed. **Why It Matters** - **Complete**: Branch and bound is complete — given enough time, it will always find the answer. - **Efficient Pruning**: Smart branching heuristics and tight bounds dramatically reduce the search space. - **α,β-CROWN**: State-of-the-art tools (winners of VNN-COMP) combine GPU-accelerated bound propagation with branch-and-bound. **Branch and Bound** is **divide and conquer for verification** — recursively splitting the problem until every subregion is proven safe or a counterexample is found.