ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 1, pp. 24–38.
Pleiades Publishing, Inc., 2016.
Original Russian Text
V.D. Shmatkov, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 1, pp. 27–42.
COMMUNICATION NETWORK THEORY
Lattice Flows in Networks
V. D. Shmatkov
Ryazan State Radio Engineering University, Ryazan, Russia
Received June 2, 2014; in ﬁnal form, October 1, 2015
Abstract—We consider ﬂows in networks analogous to numerical ﬂows but such that values of
arc capacities are elements of a lattice. We present an analog of the max-ﬂow min-cut theorem.
However, ﬁnding the value of the maximum ﬂow for lattice ﬂows is based on not this theorem
but computations in the algebra of matrices over the lattice; in particular, the maximum ﬂow
value is found with the help of transitive closure of ﬂow capacity functions. We show that there
exists a correspondence between ﬂows and solutions of special-form systems of linear equations
over distributive lattices.
Classical numerical ﬂows in networks are well studied (see, e.g., [1, 2]). A key result in the
network ﬂow theory is the theorem on equality of values of the maximum ﬂow and minimum cut.
Note that some directions of further development of the ﬂow theory are related to the lattice
theory approach. Thus, for example, using the observation that the set of all cuts forms a lattice,
numerous algorithms for ﬁnding minimum cuts are constructed [3,4]. These algorithms are based
on submodular function minimization theory . Matroid theory is closely related with lattices;
for instance, matroid ﬂats form a geometric lattice . In  it is shown that the max-ﬂow min-
cut theorem can be formulated as a statement on weighted cycles and cocycles in matroids. An
extension of the max-ﬂow min-cut theorem to the case of multicommodity ﬂows is considered
in . In , the max-ﬂow min-cut theorem in mutlicomodity ﬂows was considered for matroids.
In Proposition 3 of the present paper, the set of ﬂows is lattice ordered in a natural way. Numerical
ﬂows were lattice ordered in . The max-ﬂow min-cut theorem was studied from the topology
viewpoint in , where an analog of Theorem 4 for Boolean lattices was also presented.
Examples of ﬂows considered in the present paper can be information ﬂows between servers,
data ﬂows, ﬁle storage model, commodity ﬂows, and neural networks.
In Theorem 1 we show that every ﬂow is a sum of ﬂows along paths. In Theorem 2 we establish
that for an acyclic relation the ﬂow value equals the ﬂow value over any cut. In Theorem 4 we
show that for an acyclic relation the maximum ﬂow value is equal to the meet of capacities of all
cuts, and in Theorem 6 we establish that the maximum ﬂow value equals the join of the transitive
closure of the capacity function from the source to the sink with the trace of the transitive closure
of the capacity functions. In Theorem 7 we present formulas for computing the maximum ﬂow.
Note that by Theorem 5 the value of the transitive closure of matrices over a distributive
lattice equals the value of the meet of cut values. Theorem 5 generalizes Theorem 1 in [12, ch. 8,
Note that in the literature there were considered other generalizations of numerical ﬂows, for
instance, multicommodity ﬂows (see ).