The perfect matching problem has a randomized $NC$ algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph, ensures that it has a unique minimum weight perfect matching, with a good probability. We ... more >>>

We show that the bipartite matching problem is in NC. We extend the result to weighted bipartite matching and the computation of the noncommutative rank of a symbolic matrix. In particular, this implies that the decision version of linear matroid intersection is in NC as well. The techniques are based ... more >>>

The recent paper \cite{chatterjee2026bipartite} showed that deciding whether a bipartite graph has a perfect matching can be reduced to deciding whether a determinant, whose value may be assigned to any sufficiently large field $\mathbb{F}$, equals to zero. In the second part of their work, \cite{chatterjee2026bipartite} also generalized the algebraic method ... more >>>