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TR19-021 | 19th February 2019
Rahul Ilango

#### $AC^0[p]$ Lower Bounds and NP-Hardness for Variants of MCSP

The Minimum Circuit Size Problem (MCSP) asks whether a (given) Boolean function has a circuit of at most a (given) size. Despite over a half-century of study, we know relatively little about the computational complexity of MCSP. We do know that questions about the complexity of MCSP have significant ramifications ... more >>>

TR19-020 | 4th February 2019
Ludmila Glinskih, Dmitry Itsykson

#### On Tseitin formulas, read-once branching programs and treewidth

We show that any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on an $n\times n$ grid graph has size at least $2^{\Omega(n)}$. Then using the Excluded Grid Theorem by Robertson and Seymour we show that for arbitrary graph $G(V,E)$ any nondeterministic read-once branching program that computes ... more >>>

TR19-019 | 19th February 2019
Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi

#### Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\mathop{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas (MFCS ... more >>>

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