Alternation-Trading Proofs, Linear Programming, and Lower Bounds

TL;DR

This paper introduces a linear programming-based framework to analyze and automate alternation-trading proofs, leading to new lower bounds for computational problems and insights into the limitations of current proof techniques.

Contribution

It develops a methodology that formalizes alternation-trading proofs, enabling automated analysis and deriving new lower bounds for natural computational problems.

Findings

Linear programming can model the search for lower bounds.

Automated theorem proving yields new human-readable lower bounds.

Framework identifies limitations of existing proof techniques.

Abstract

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proof-by-contradiction strategy that we call alternation-trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on this result, we extract new human-readable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques.