Alternation-Trading Proofs, Linear Programming, and Lower Bounds

  • Authors:

  • Ryan Williams

  • Affiliations:

  • Stanford University

  • Venue:
  • ACM Transactions on Computation Theory (TOCT)
  • Year:
  • 2013

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Abstract

A fertile area of recent research has demonstrated concrete polynomial-time lower bounds for 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 proof-by-contradiction strategy that we call resource trading or 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 these results, we extract new human-readable time lower bounds for several problems and identify patterns that allow for further generalization. The framework can also be used to prove concrete limitations on the current techniques.