Tight Upper Bound on Splitting by Linear Combinations for Pigeonhole Principle

Abstract

The usual DPLL algorithm uses splittings (branchings) on single Boolean variables. We consider an extension to allow splitting on linear combinations mod 2, which yields a search tree called a linear splitting tree. We prove that the pigeonhole principle has linear splitting trees of size $2^{O(n)}$. This is near-optimal since Itsykson and Sokolov [1] proved a $2^{\varOmega (n)}$ lower bound. It improves on the size $2^{\varTheta (n \log n)}$ for splitting on single variables; thus the pigeonhole principle has a gap between linear splitting and the usual splitting on single variables. This is of particular interest since the pigeonhole principle is not based on linear constraints. We further prove that the perfect matching principle has splitting trees of size $2^{O(n)}$.

Research is partially supported by the Government of the Russian Federation under Grant 14.Z50.31.0030.