above lower bound of Erdös and Pósa9 on the number of
Let us repeat the procedure of eliminating a maximum
size set of edge-disjoint cycles. It follows from the lemma
that after O(log2 n) rounds, each component of the obtained
graph will have a constant difference between the number of
edges and vertices. At this stage, each component will have
only constantly many cycles. And so, in one more round we
will eliminate all cycles.
A different view on the third approach is by considering
the dimension of the perfect matching polytope. For a connected bipartite graph, where each of its edges belong to
some perfect matching, the perfect matching polytope has
dimension m − n + 1 [Lovász and Plummer,
20 Theorem 7. 6. 2].
Thus, the argument of this approach can also be viewed as
decreasing the dimension of the perfect matching polytope
by a fraction in each round and eventually reaching dimension zero, that is, just one perfect matching point.
4. FURTHER DEVELOPMENTS
After years of inactivity, our result inspired a series of follow-up works on parallel algorithms for perfect matching and the
Isolation Lemma. In one direction, our isolation approach
was generalized to the broader settings of matroid intersection and polytopes with totally unimodular faces, respectively.
15, 16 For these general settings, the right substitute for
cycles are integer vectors parallel to a face of the associated
polytope. Following our first approach, if one eliminates
vectors of length ≤2i, then there are only polynomially many
vectors of length ≤2i+ 1, in their respective settings (see15, 16
for details). It is not clear, however, if our second and third
approaches work in these settings.
In another direction, Svensson and Tarnawski24 generalized the isolation result to perfect matchings in general
graphs. They use the basic framework of our first approach as
the starting point, but they need to combine the technique
of eliminating cycles with a second parameter (
contractabil-ity) to control the progress in subsequent rounds.
The techniques developed by us and by Svensson and
Tarnawski were used by Anari and Vazirani3 to compute a
perfect matching in planar graphs in NC (see also22), which
also was a long-standing open problem. They show that the
sets to be contracted, odd sets of vertices that form a tight
cut in the LP-constraints, can be computed in NC. In a subsequent work, the NC algorithm was further generalized to
The Isolation Lemma has applications in many different
settings—in particular, in design of randomized algorithms.
The main open question that remains is for what other settings
can one derandomize the Isolation Lemma. We conjecture
that our isolation approach works for any family of sets whose
corresponding polytopes are described by 0/1 constraints.
We would like to thank Manindra Agrawal and Nitin Saxena
for their constant encouragement and very helpful discussions. We thank Arpita Korwar for discussions on some
other techniques used in this research, Jacobo Torán for
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discussions on the number of shortest cycles, and Nisheeth
Vishnoi for helpful comments.
Stephen Fenner ( email@example.com),
University of South Carolina, Columbia,
Rohit Gurjar (rohitgurjar0@gmail.
com), California Institute of Technology,
Pasadena, CA, USA.
Thomas Thierauf (thomas.
firstname.lastname@example.org), Aalen University,
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