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Graphs, Networks and Algorithms pp 219–249Cite as

Combinatorial Applications

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Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 5))

Abstract

In this chapter, we use the theorems of Ford and Fulkerson about maximal ows to prove some central results in combinatorics. More precisely, we will use ow theory to study disjoint paths in graphs, matchings in bipartite graphs, transversals of set families, the combinatorics of matrices, partitions of directed graphs, partially ordered sets, parallelisms of complete designs, and the supply and demand theorem. In particular, transversal theory can be developed from the theory of ows on networks; this approach was first suggested in 1962 in the book by Ford and Fulkerson. Compared with the usual alternative approach of taking Philip Hall’s marriage theorem—which we will treat in Sect. 7.3—as the starting point of transversal theory, this way of proceeding has a distinct advantage: it also yields algorithms allowing explicit constructions for the objects in question.

Everything flows.

Heraclitus

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Notes

  1. 1.

    For more on network reliability, we recommend [Col87].

  2. 2.

    Quite often, this result is stated in the language of matrices instead; see Theorem 7.4.1 below.

  3. 3.

    This theorem is likewise often stated in different language; see Theorem 7.3.1.

  4. 4.

    For an intuitive interpretation, we might think of the A i as certain groups of people who each send a representative a i into a committee. Then the SDR property means that no committee member is allowed to represent more than one group, and the transversal {a 1,…,a n } just is the committee. Another interpretation will be given below, after Theorem 7.3.1.

  5. 5.

    Of course, we might as well choose the edges in C∩M T .

  6. 6.

    A strong generalization of Theorem 7.4.5 is proved in [LewLL86].

  7. 7.

    Note that this function differs from the determinant of A only by the signs of the terms appearing in the sum. Although there exist efficient algorithms for computing determinants, evaluating the permanent of a matrix is NP-hard by a celebrated result of Valiant [Val79a].

  8. 8.

    This problem was generalized by Gabow and Tarjan, [GabTa88] who also gave an algorithm with complexity O((|V|log|V|)1/2|E|). For our classical special case, this yields a complexity of O((log|V|)1/2|V|5/2).

  9. 9.

    Note that independent sets are the vertex analogue of matchings, which may be viewed as independent sets of edges; hence the notation α′(G) in Sect. 7.2 for the maximal cardinality of a matching.

  10. 10.

    However, the reverse inequality does not hold for this more general case, as the example in Fig. 7.3 shows.

  11. 11.

    The term tournament becomes clear by considering a competition where there are no draws: for example, tennis. Assume that each of n players (or teams, as the case may be) plays against every other one, and that the edge {i,j} is oriented as ij if i wins against j. Then an orientation of K n indeed represents the outcome of a (complete) tournament.

  12. 12.

    A somewhat more general problem will be the subject of Chap. 11.

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    Dieter Jungnickel

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Jungnickel, D. (2013). Combinatorial Applications. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_7

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