On Lower Bound Methods for Tree-like Cutting Plane Proofs

TL;DR

This paper improves lower bound techniques for Tree-like Cutting Plane proofs by providing an explicit randomized protocol for the GreaterThan function, linking KW game bounds to proof size bounds.

Contribution

It presents a full O(log n + log(1/epsilon))-bit randomized protocol for GreaterThan based on noisy binary search, connecting KW game bounds to Tree-CP proof size lower bounds.

Findings

Established a new explicit randomized protocol for GreaterThan.

Linked randomized KW game bounds to Tree-CP proof size bounds.

Demonstrated the equivalence of randomness and coefficient size in lower bounds.

Abstract

In the book Boolean Function Complexity by Stasys Jukna, two lower bound techniques for Tree-like Cutting Plane proofs (henceforth, "Tree-CP proofs") using Karchmer-Widgerson type communication games (henceforth, "KW games") are presented: The first, applicable to Tree-CP proofs with bounded coefficients, translates Omega(t) deterministic lower bounds on KW games to 2^Omega(t/log n) lower bounds on Tree-CP proof size. The second, applicable to Tree-CP proofs with unbounded coefficients, translates Omega(t) randomized lower bounds on KW games to 2^Omega(t/log^2 n) lower bounds on Tree-CP proof size. The textbook proof in the latter case uses a O(log^2 n)-bit randomized protocol for the GreaterThan function. However, Nisan mentioned using the ideas of Feige, et al. to construct a O(log n + log(1/epsilon))-bit randomized protocol for GreaterThan. Nisan did not explicitly give the proof, though later results in his paper assume such a protocol. In this short exposition, we present the full O(log n + log(1/epsilon))-bit randomized protocol for the GreaterThan function based on the ideas of Feige, et al. for "noisy binary search." As an application, we show how to translate Omega(t) randomized lower bounds on KW games to 2^Omega(t/log n) lower bounds on Tree-CP proof size in the unbounded coefficient case. This equates randomness with coefficient size for the Tree-CP/KW game lower bound method. We believe that, while the O(log n + log(1/epsilon))-bit randomized protocol for GreaterThan is a "known" result, the explicit connection to Tree-CP proof size lower bounds given here is new.