The deficiency of a propositional formula F in CNF with n variables
and m clauses is defined as m-n. It is known that minimal
unsatisfiable formulas (unsatisfiable formulas which become
satisfiable by removing any clause) have positive deficiency.
Recognition of minimal unsatisfiable formulas is NP-hard, and it was
shown recently ... more >>>

We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size and $O(\log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth.

We obtain our result by an almost complete ... more >>>

In this paper we present a pseudo-deterministic $RNC$ algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses $poly(n)$ processors, $poly({\log n})$ depth, $poly(\log n)$ random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That ... more >>>

The recent breakthrough work of Chatterjee, Ghosh, Gurjar, Raj and Thierauf [CGGRT26] gives the first deterministic NC algorithm for the bipartite matching problem. They show how to detect as well as find perfect matchings in bipartite graphs in NC. In this note we present an arguably simpler-to-state variation of the ... more >>>