Lifting Theorems Meet Information Complexity: Known and New Lower Bounds of Set-disjointness

TL;DR

This paper introduces a new density increment method combining information complexity and lifting theorems to establish improved lower bounds for set-disjointness problems in communication complexity.

Contribution

It presents a simplified, unified proof technique for lower bounds in multi-party set-disjointness, improving previous bounds and offering new insights through a density increment approach.

Findings

Proves large rectangles cannot be 0-monochromatic for multi-party unique-disjointness.

Provides an alternative proof of randomized communication lower bounds.

Simplifies and enhances bounds for sparse unique-disjointness.

Abstract

Set-disjointness problems are one of the most fundamental problems in communication complexity and have been extensively studied in past decades. Given its importance, many lower bound techniques were introduced to prove communication lower bounds of set-disjointness. Combining ideas from information complexity and query-to-communication lifting theorems, we introduce a density increment argument to prove communication lower bounds for set-disjointness: We give a simple proof showing that a large rectangle cannot be $0$-monochromatic for multi-party unique-disjointness. We interpret the direct-sum argument as a density increment process and give an alternative proof of randomized communication lower bounds for multi-party unique-disjointness. Avoiding full simulations in lifting theorems, we simplify and improve communication lower bounds for sparse unique-disjointness. Potential applications to be unified and improved by our density increment argument are also discussed.