Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most $k$ is Zariski-closed, an important property in ... more >>>

Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk (Proc. of the 49th Annual {ACM} {SIGACT} Symposium on Theory of Computing (STOC), pages {653--664}, 2017) and independently by Grochow, Kumar, Saks and Saraf~(CoRR, abs/1701.01717, 2017) as an attempt to transfer Razborov and Rudich's famous barrier result (J. Comput. ... more >>>

We consider the problem of commutative rank computation of a given matrix space, $\mathcal{B}\subseteq\mathbb{F}^{n\times n}$. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is $n$, subsumes problems such as testing perfect matching in graphs ... more >>>

We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that $A$ is fixed. We obtain the following dichotomy: If $A/rad(A)$ is noncommutative, then computing the determinant over $A$ is hard. ``Hard'' here means $\#P$-hard over fields of characteristic $0$ and $ModP_p$-hard over ... more >>>

We study private computations in information-theoretical settings on
networks that are not 2-connected. Non-2-connected networks are
``non-private'' in the sense that most functions cannot privately be
computed on such networks. We relax the notion of privacy by
introducing lossy private protocols, which generalize private
protocols. We measure the information each ... more >>>

We study the role of connectivity of communication networks in private
computations under information theoretic settings. It will be shown that
some functions can be computed by private protocols even if the
underlying network is 1-connected but not 2-connected. Then we give a
complete characterisation of non-degenerate functions that can ... more >>>