A complete resolution of the Keller maximum clique problem

  • Authors:

  • Jennifer Debroni;John D. Eblen;Michael A. Langston;Wendy Myrvold;Peter Shor;Dinesh Weerapurage

  • Affiliations:

  • University of Victoria, Victoria, B. C. Canada;University of Tennessee, Knoxville, TN;University of Tennessee, Knoxville, TN;University of Victoria, Victoria, B. C. Canada;Massachusetts Institute of Technology, Cambridge, MA;University of Tennessee, Knoxville, TN

  • Venue:
  • Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2011

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Abstract

A d-dimensional Keller graph has vertices which are numbered with each of the 4d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a "high speed computer the size of a major galaxy". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.