The resolution complexity of random graphk-colorability

  • Authors:

  • Paul Beame;Joseph Culberson;David Mitchell;Cristopher Moore

  • Affiliations:

  • Computer Science and Engineering, University of Washington, Seattle, WA;Department of Computer Science, University of Alberta, Edmonton, Alta, Canada;Department of Computer Science, Simon Fraser University, Burnaby, BC, Canada;Computer Science Department, University of New Mexico, Albuquerque, NM

  • Venue:
  • Discrete Applied Mathematics - Special issue: Typical case complexity and phase transitions
  • Year:
  • 2005

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Abstract

We consider the resolution proof complexity of propositional formulas which encode random instances of graph k-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linear-exponential lower bounds on the size of resolution refutations. For random graphs with n vertices and any ε 0, we obtain a lower-bound tradeoff between graph density and refutation size that implies subexponential lower bounds of the form 2nδ for some δ 0 for non-k-colorability proofs of graphs with n vertices and O(n3/2-1/k-ε) edges. We obtain sharper lower bounds for Davis-Putnam-DPLL proofs and for proofs in a system considered by McDiarmid.These proof complexity bounds imply that many natural algorithms for k-coloring or k-colorability have essentially the same exponential tradeoff lower bounds on their running times. We also show that very simple algorithms for k-colorability have upper bounds on their running times that are qualitatively similar to the lower bounds as a function of the graph density.