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Revision #1 to TR17-151 | 18th May 2022 19:52

#### Stabbing Planes

Revision #1
Authors: Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere
Accepted on: 18th May 2022 19:52
Keywords:

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 $ax \leq b$, 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 R(CP) 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.

### Paper:

TR17-151 | 8th October 2017 17:36

#### Stabbing Planes

TR17-151
Authors: Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere
Publication: 9th October 2017 02:20