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  1. SIAM Journal on Discrete Mathematics (SJDMEC)
  2. Volume 8
  3. Volume 8 Issue 3
  4. The Complexity of the Hajs Calculus
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Volume 8
Volume 8 Issue 4
Volume 8 Issue 3
The Complexity of the Hajs Calculus
P-Components and the Homogeneous Decomposition of Graphs
Multidimensional Divide-and-Conquer Maximin Recurrences
Approximation Algorithms for Minimum-Time Broadcast
Combinatorial Algorithm for a Lower Bound on Frame Rigidity
Tree Spanners
An Inequality for Probability Moments with Applications
Volume 8 Issue 2
Volume 8 Issue 1
Volume 7
Volume 6
Volume 5
Volume 4
Volume 3
Volume 2
Volume 1

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The Complexity of the Hajs Calculus

Content Provider Society for Industrial and Applied Mathematics (SIAM)
Author Pitassi, Toniann Urquhart, Alasdair
Copyright Year 1995
Abstract The Hajs calculus is a simple, nondeterministic procedure that generates the class of non-3-colorable graphs. Mansfield and Welsch posed the question of whether there exist graphs that require exponential-sized Hajs constructions. Unless ${\text{NP}} \ne {\text{coNP}}$, there must exist graphs that require exponential-sized constructions, but to date, little progress has been made on this question, despite considerable effort. In this paper, we prove that the Hajs calculus generates polynomial-sized constructions for all non-3-colorable graphs if and only if extended Frege systems are polynomially bounded. Extended Frege systems are a very powerful family of proof systems for proving tautologies, and proving superpolynomial lower bounds for these systems is a long-standing, important problem in logic and complexity theory. We also establish a relationship between a complete subsystem of the Hajs calculus and bounded-depth Frege systems; this enables us to prove exponential lower bounds on this subsystem of the Hajs calculus.
Starting Page 464
Ending Page 483
Page Count 20
File Format PDF
ISSN 08954801
DOI 10.1137/S089548019224024X
e-ISSN 10957146
Journal SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number 3
Volume Number 8
Language English
Publisher Society for Industrial and Applied Mathematics
Publisher Date 2006-08-01
Access Restriction Subscribed
Subject Keyword 3-colorability complexity of propositional proof systems Complexity of proofs graph constructions Graph theory Graph algorithms
Content Type Text
Resource Type Article
Subject Mathematics
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