Stabbing Planes

TL;DR

Stabbing Planes is a new proof system modeling modern branch-and-cut algorithms for integer programming, capable of simulating cutting plane proofs, with proven lower bounds on proof complexity and communication complexity.

Contribution

It introduces Stabbing Planes, a semi-algebraic proof system that formalizes branch-and-cut methods and demonstrates equivalence to certain proof systems, with new lower bounds on proof and communication complexity.

Findings

Stabbing Planes can simulate Cutting Planes proofs efficiently.

It is equivalent to a tree-like RCP system.

Linear lower bounds on proof rank and unbalanced proofs are established.

Abstract

We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers. As with DPLL there is only a single rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane $ax \geq b$ with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying $ax \geq b$, the points satisfying $ax \leq b-1$, and the middle slab $b - 1 < ax < b$. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show that Stabbing Planes can efficiently simulate the Cutting Planes proof system, and is equivalent to a tree-like variant of the RCP system of [Krajicek98]. As well, we show that it possesses short proofs of the canonical family of systems of $\mathbb{F}_2$-linear equations known as the Tseitin formulas. Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot be balanced. In doing so, we show that real communication protocols cannot be balanced and establish the first lower bound on the real communication complexity of the set disjointness function.