Two Remarks Concerning Balanced Matroids

The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of “sparse” paving matroids that are balanced, but at the same time rich enough combinatorially to permit the encoding of hard counting problems; and (b) a paving matroid that is not balanced. The computational significance of (a) is the following. As a consequence of balance, there is an efficient algorithm for approximating the number of bases of a sparse paving matroid within specified relative error. On the other hand, determining the number of bases exactly is likely to be computationally intractable.

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Correspondence to Mark Jerrum*.

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* The work described here was supported by the grant “Sharper analysis of randomised algorithms” from the UK Engineering and Physical Sciences Research Council.

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Jerrum*, M. Two Remarks Concerning Balanced Matroids. Combinatorica 26, 733–742 (2006). https://doi.org/10.1007/s00493-006-0039-5

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Mathematics Subject Classification (2000):

  • 05B35
  • 05A99
  • 51E10
  • 68Q17