ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 4, pp. 356–370.
Pleiades Publishing, Inc., 2006.
Original Russian Text
M.A. Babenko, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 4, pp. 104–120.
COMMUNICATION NETWORK THEORY
On Flows in Simple Bidirected and
M. A. Babenko
Lomonosov Moscow State University, Moscow
Received June 1, 2006; in ﬁnal form, August 8, 2006
Abstract—We consider a simple directed network. Results of Karzanov, Even, and Tarjan
show that the blocking ﬂow method constructs a maximum integer ﬂow in this network in
)) time (hereinafter, n denotes the number of nodes, and m the number of
arcs or edges). For the bidirected case, Gabow proposed a reduction to solve the maximum
integer ﬂow problem in O(m
) time. We show that, with a variant of the blocking ﬂow method,
this problem can also be solved in O(mn
) time. Hence, the gap between the complexities
of directed and bidirected cases is eliminated. Our results are described in the equivalent
terms of skew-symmetric networks. To obtain the time bound of O(mn
), we prove that
the value of an integer s–s
ﬂow in a skew-symmetric network without parallel arcs does not
), where d is the length of the shortest regular s–s
path in the split network
and U is the maximum arc capacity. We also show that any acyclic integer ﬂow of value v in a
skew-symmetric network without parallel arcs can be positive on at most O(n
Network ﬂow theory is a widely known and deeply studied branch of combinatorial optimization.
A network ﬂow corresponds to the process that has a natural physical model; namely, the goal is to
transfer the maximum amount of substance from one node of a directed graph to another, obeying
capacity constraints and without losses in intermediate nodes. It turns out possible to describe
a number of applied optimization, scheduling, and information transmission problems in terms
of network ﬂows (for details, see ). Moreover, some purely combinatorial problems (examples
include s–t path packing, matching in bipartite graphs, packing of chains in partially ordered
sets, etc.) can be reduced to the maximum ﬂow problem . This way, ﬂow theory becomes
a convenient generalizing tool, and maximum ﬂow algorithms get a large number of practical
However, there is a wide class of tasks that indicate a certain level of analogy with classical
network ﬂow problems but still cannot be adequately expressed and solved in these terms. A typical
example is the problem of ﬁnding maximum matching in the general (not necessarily bipartite)
graph. For these problems, the notions of bidirected graphs and bidirected ﬂows proved to be useful.
These notions were earlier introduced by Edmonds and Johnson in connection with an important
class of integer programs [2–4]. For bidirected graphs there exists an alternative and somewhat
more convenient language of skew-symmetric graphs. The latter notion was introduced by Goldberg
and Karzanov [5, 6], and also by Tutte  (where these graphs were called antisymmetrical).
Supported in part by the Russian Foundation for Basic Research, project nos. 03-01-00475 and 06-01-
00122. The work was partly done while the author was visiting Microsoft Research Corp.