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

We study the complexity of cutting planes and branching schemes from a theoretical point of view. We give some rigorous underpinnings to the empirically observed phenomenon that combining cutting planes and branching into a branch-and-cut framework can be orders of magnitude more efficient than employing these tools on their own. In particular, we give general conditions under which a cutting plane strategy and a branching scheme give a provably exponential advantage in efficiency when combined into branch-and-cut. The efficiency of these algorithms is evaluated using two concrete measures: number of iterations and sparsity of constraints used in the intermediate linear/convex programs. To the best of our knowledge, our results are the first mathematically rigorous demonstration of the superiority of branch-and-cut over pure cutting planes and pure branch-and-bound.

💡 Research Summary

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This paper investigates the theoretical complexity of cutting‑plane methods and branching schemes in mixed‑integer optimization (MIO) and provides the first rigorous proof that a combined branch‑and‑cut framework can be exponentially more efficient than using either tool alone. The authors formalize cutting‑plane paradigms (CP) as functions that, given a closed convex set C, generate families of valid cuts, and branching schemes (D) as families of disjunctions that partition the integer lattice. They introduce a sparsity parameter s, limiting the number of non‑zero coefficients in the linear description of each cut or each polyhedron of a disjunction. This reflects the practical concern that solvers prefer sparse constraints for memory and computational efficiency.

The algorithmic model is a non‑deterministic proof system: each node of the branch‑and‑cut tree corresponds to a convex relaxation that must be solved (an LP or more general convex program). The size of the tree equals the number of such relaxations, providing a natural measure of algorithmic effort. The authors analyze two primary performance metrics: (i) the number of LP/convex relaxations processed (tree size) and (ii) the sparsity of the constraints defining those relaxations.

The first set of results concerns pure branch‑and‑bound with split disjunctions of bounded sparsity. By adapting Jeroslow’s classic example, they show that if the sparsity s is at most ⌊n/2⌋ (i.e., only very sparse disjunctions are allowed), any branch‑and‑bound proof of optimality for the instance
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