  The maximum integer skew-symmetric flow problem (MSFP) generalizes both the
maximum flow and maximum matching problems. It was introduced by Tutte in terms
of self-conjugate flows in antisymmetrical digraphs. He showed that for these
objects there are natural analogs of classical theoretical results on usual
network flows, such as the flow decomposition, augmenting path, and max-flow
min-cut theorems. We give unified and shorter proofs for those theoretical
results.
  We then extend to MSFP the shortest augmenting path method of Edmonds and
Karp and the blocking flow method of Dinits, obtaining algorithms with similar
time bounds in general case. Moreover, in the cases of unit arc capacities and
unit ``node capacities'' the blocking skew-symmetric flow algorithm has time
bounds similar to those established in Even and Tarjan (1975) and Karzanov
(1973) for Dinits' algorithm. In particular, this implies an algorithm for
finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time,
which matches the time bound for the algorithm of Micali and Vazirani. Finally,
extending a clique compression technique of Feder and Motwani to particular
skew-symmetric graphs, we speed up the implied maximum matching algorithm to
run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound
for dense nonbipartite graphs.
  Also other theoretical and algorithmic results on skew-symmetric flows and
their applications are presented.

Combinatorics