Lower bounds for strictly fundamental cycle bases in grid graphs

  • Authors:

  • Ekkehard Köhler;Christian Liebchen;Gregor Wünsch;Romeo Rizzi

  • Affiliations:

  • Brandenburgische Technische Universität Cottbus, Institut für Mathematik, Postbox 10 13 44, D-03013 Cottbus, Germany;Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, D-10623 Berlin, Germany;Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, D-10623 Berlin, Germany;Università degli Studi di Udine, Dipartimento di Matematica ed Informatica, Via delle Scienze, 208, I-33100 Udine, Italy

  • Venue:
  • Networks - Special Issue on Trees
  • Year:
  • 2009

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Abstract

Consider the following problem: compute a spanning tree such that the sum of the lengths of its induced fundamental circuits is as small as possible. We motivate why planar square grid graphs are very relevant instances for this problem. In particular, other contributions already showed that the identification of strong lower bounds is highly challenging. Asymptotically, for a graph on n vertices, Alon et al. [SIAM J Comput 24(1995), 78–100] obtained a lower bound of Ω(n log n). We raise the n log n coefficient by a factor of 325. Concerning optimality proofs, the largest grid for which provably optimum solutions were known is 6 × 6, and it was obtained by massive MIP computing power. Here, we present a combinatorial optimality proof even for the 8 × 8 grid. These two results are complemented by new combinatorial lower bounds for the dimensions in which earlier empirical computations were performed, i.e., for up to 10,000 vertices. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009