Efficient approximation algorithms for shortest cycles in undirected graphs

  • Authors:

  • Andrzej Lingas;Eva-Marta Lundell

  • Affiliations:

  • Department of Computer Science, Lund University, Lund, Sweden;Department of Computer Science, Lund University, Lund, Sweden

  • Venue:
  • LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
  • Year:
  • 2008

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Abstract

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. For an undirected graph G of unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k + 2 for odd k, in time O(n3/2 √log n). Thus, in general, it yields a 2 2/3 approximation. We study also the problem of finding a simple cycle of minimum total weight in an undirected graph with nonnegative edge weights. We present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle in an undirected graph with nonnegative integer edge weights in the range {1, 2, ..., M}. This algorithm runs in time O(n2 log n log M).