Finding a minimum circuit in a graph

ABSTRACT

Finding minimum circuits in graphs and digraphs is discussed. An almost minimum circuit is a circuit which may have only one edge more than the minimum. An 0(n2) algorithm is presented to find an almost minimum circuit. The straightforward algorithm for finding a minimum circuit has an 0(ne) behavior. It is refined to yield an 0(n2) average time algorithm. An alternative method is to reduce the problem of finding a minimum circuit to that of finding a triangle in an auxiliary graph. Three methods for finding a triangle in a graph are presented. The first has an 0(e3/2) worst case bound ((n) for planar graphs); the second takes 0(n5/3) time on the average; the third has an 0(nlog7) worst case behavior. For digraphs, recent results of Bloniarz, Fisher and Meyer are used to obtain an algorithm with 0(n2logn) average behavior.

REFERENCES

• 1: Alfred V. Aho , John E. Hopcroft, The Design and Analysis of Computer Algorithms, Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1974
• 2: P.A.Bloniarz, M.J. Fisher and A.R. Meyer, "A Note on the Average Time to Compute Transitive Closures", Proc. of the 3rd Int. Colloquium on Automata, Languages and Programming, S. Michelson and R. Milner (eds.),July 1976.
• 3: P. Erdös and J. Spencer, "Probabilistic Methods in Combinatorics", Academic Press, 1974.
• 4: C.L.Liu, "Introduction to Combinatorial Mathematics", McGraw-Hill, 1968.
• 5: "Gaussian Elimination is not Optimal", Numerische Mathematik 13, 354-356.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Path and circuit problems

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Trees

General Terms:
Algorithms, Theory