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Circulations

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Graphs, Networks and Algorithms

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

Abstract

In Chap. 6, we considered the simplest kind of ow problems, namely the determination of maximal ows in a network, in considerable detail; and in Chap. 7, we studied various important applications of this theory. The present chapter deals with generalizations of the

ows we worked with so far. For example, quite often there are also lower bounds on the capacities of the edges given; or one may seek a maximal ow which is optimal with respect to a given cost function on the edges. To solve these (and even more general) problems, it proves helpful to remove the exceptional role of the source and sink vertices s and t by requiring the ow preservation condition for all vertices, including s and t. This leads to the notion of circulations on directed graphs. As we will see, there are many interesting applications of this more general concept. To a large part, these cannot be treated using the theory of maximal ows as presented before; nevertheless, the methods of Chap. 6 will serve as fundamental tools for the more general setting. We shall begin with a rather thorough theoretical investigation of circulations and then develop efficient algorithms for finding an optimal circulation (or showing that no such circulation can exist); this turns out to be considerably more difficult than determining an ordinary maximal ow. Finally, as an application of circulations, we shall also consider the construction of rather good codes (a mathematical concept used to deal with data corruption by allowing the correction of errors) using the cycle spaces of graphs.

Round and round the circle…

T. S. Eliot

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Notes

  1. 1.

    We may assume, without loss of generality, that ts is not an edge of G; otherwise, we could subdivide ts into two edges.

  2. 2.

    As usual, ∞ stands for a sufficiently large number, for example \(\sum_{e^{-}=s} c(e)\).

  3. 3.

    The requirement b(r)=0 guarantees that only flows with a non-negative value correspond to feasible circulations. Note that we need to add the condition w(f)≥0 to the definition in terms of flows, as trivial examples show. We certainly do not want to consider flows with a negative value, as these would (intuitively) correspond to a flow in the reverse direction, namely from t to s.

  4. 4.

    The cycle space of a graph is a special case of a more general construction assigning certain modules to any geometry; see [GhiJu90] and the references given there.

  5. 5.

    We note that the out-of-kilter algorithm is polynomial if we include the capacities in the calculation of the size of the input data: for a natural number z, we may take log2 z as a measure for the size of z. Algorithms which are polynomial in our sense (that is, their complexity is independent of the capacity constraints and the cost function) are often called strongly polynomial in the literature. For this property, all numbers occurring during the algorithm have to be polynomial in the total size of the input data. This is trivially true if the algorithm involves only additions, subtractions, comparisons, and multiplications or divisions by a constant factor. We will sometimes call algorithms which are polynomial in |E|, |V|, and the logarithm of the size of the input data weakly polynomial.

  6. 6.

    A similar argument shows that we do not need parallel edges; indeed, we have always excluded parallel edges in our study of flows and circulations.

  7. 7.

    Of course, c is not completely arbitrary: we should have −c(v,u)≤c(u,v); otherwise, there are no feasible circulations on N.

  8. 8.

    Note that the transformation we have described tacitly assumes that pairs of antiparallel edges in the original digraph have the same cost; otherwise, the elimination technique used makes no sense. While this assumption will be satisfied in many applications, we can use the following little trick if it should be violated: for each pair of antiparallel edges with different costs, we subdivide one of the edges into two edges and then assign to both new edges the same capacities and half the cost of the original edge.

  9. 9.

    For an intuitive interpretation of a circulation in the new sense, consider only its positive part—which lives on a network without antiparallel edges.

  10. 10.

    Note that the following transformation has the same form as the one used in Eq. (10.7).

  11. 11.

    This optimality criterion for potentials is one example for how the different way of description used in this section simplifies the technical details of our presentation. Of course, an analogous criterion can be proved using the standard notation for a network (G,b,c). However, we would then need three conditions which have to be satisfied for all edges uv∈E:

    see [AhuMO93], p. 330. These conditions are called complementary slackness conditions; they are a special case of the corresponding conditions used in linear programming. It may be checked that the potentials p(v) correspond to the dual variables if the problem of determining an optimal circulation is written as a linear program.

  12. 12.

    In our context, the terms mean cost or mean length would make more sense; however, we do not want to deviate from common usage.

  13. 13.

    Sometimes, there are upper bounds placed also on the capacities of the edges.

  14. 14.

    The author of the present book thinks that the most intuitive way to become acquainted with problems of combinatorial optimization is the presentation in a graph theory context; however, the theory of linear programming is indispensable for further study.

  15. 15.

    [Gro85, §8.2] shows an actual example where a chain of transactions resulting in a gain occurs.

  16. 16.

    In everyday language, a code usually means a way of secret communication: for instance, one speaks of breaking a code. However, the corresponding part of mathematics is cryptography, whereas coding theory deals with the problem of ensuring that information can be recovered accurately after transmission or storage, in spite of possible loss or corruption of data; there is no aspect of secrecy involved. To use the words of Claude Shannon [Sha48a, Sha48b], the founder of information theory (who, by the way, also put cryptography onto a sound mathematical basis in his paper [Sha49b]), of which coding theory is one part:

    The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.

    Codes in this sense constitute a spectacularly successful part of applied mathematics: satellite images, CD’s, DVD’s, digital TV, mobile phones all need codes (to mention just a few examples). In this context, we also recommend another introduction to coding theory, namely [VanOo89], where the reader may find a nice description of the mathematical principles underlying compact discs.

  17. 17.

    This is precisely the approach used for transmitting pictures from one of the early satellites, namely Mariner 9. The injection chosen in this case made it possible to correct up to seven errors per codeword, which was sufficient to make an incorrect decoding very unlikely; see [Pos69] for details.

  18. 18.

    Note that we may reconstruct x uniquely from y and the error pattern E, provided that we are in the binary case.

  19. 19.

    Finite fields are an important tool in many applications, not just in coding theory, but also in cryptography, signal processing, geometry, and design theory, as well as in algebraic graph theory and matroid theory, to mention just some examples. It is known that the cardinality of a finite field has to be a prime power q, and that there is—up to isomorphism—exactly one finite field with q elements (for every prime power q), which is usually denoted by GF(q). For background on finite fields, we refer the reader to the standard text book [LidNi94] or to [McEl87] for an introduction requiring very little algebraic knowledge; both of these books also discuss some interesting applications.

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  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Dieter Jungnickel

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Jungnickel, D. (2013). Circulations. 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_10

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