319 research outputs found
Maximum Skew-Symmetric Flows and Matchings
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 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 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.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor
stylistic corrections and shortenings to the original versio
The purity of set-systems related to Grassmann necklaces
Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky
introduced in 1998 the notion of weakly separated sets in . Moreover, they raised several conjectures on the purity for this
symmetric relation, in particular, on the Boolean cube . In
0909.1423[math.CO] we proved these purity conjectures for the Boolean cube
, the discrete Grassmanian , and some other
set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for
weakly separated collections inside a positroid which contain a Grassmann
necklace defining the positroid. We denote such set-systems as
. In this paper we give an alternative (and
shorter) proof of the purity of and present a
stronger result. More precisely, we introduce a set-system
complementary to , in
a sense, and establish its purity. Moreover, we prove (Theorem~3) that these
two set-systems are weakly separated from each other. As a consequence of
Theorem~3, we obtain the purity of set-systems related to pairs of weakly
separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise
a conjecture on the purity of both the interior and exterior of a generalized
necklace.Comment: 13 pages, 3 figure
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