## Multi-Clique-Width, a Powerful New Width Parameter

Multi-clique-width is a width parameter for graphs. It is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the size of a maximum independent set (or even the independent set polynomial) or computing the chromatic number. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs.

Tree-width and clique-width will be introduced.

Based on the ITCS 2017 paper: "Multi-Clique-Width."

## Fast and Space-Optimal Differentially-Private Low-Rank Factorization in the General Turnstile Update Model

The problem of ${\em low-rank factorization}$ of an mxn matrix A requires outputting a singular value decomposition: an $m x k$ matrix $U$, an $n x k$ matrix $V$, and a $k x k$ diagonal matrix $D$) such that $U D V^T$ approximates the matrix $A$ in the Frobenius norm. In this talk, we study releasing differentially-private low-rank factorization of a matrix in the general turnstile update model. We give a differentially-private algorithm instantiated with respect to a privacy level stronger than privacy levels for this and related problems studied in previous works, namely that of Blocki ${\text{\it et al.}}$ (FOCS 2012), Dwork ${\text{\it et al.}}$ (STOC 2014), Hardt and Roth (STOC 2012, STOC 2013), and Hardt and Price (NIPS 2014). We consider two matrices A and A' as neighboring if A - A' can be represented as an outer product of two unit vectors. Our private algorithm with respect to this privacy level incurs optimal additive error. We also prove a lower bound that shows that the space required by this algorithm is optimal up to a logarithmic factor.

Based on the paper arXiv:1604.01429
.

## Quantum Lovasz Local Lemma

If each event occurs with probability less than 1 and all events are mutually independent, then, with positive probability, none of the events will happen. Lovasz Local Lemma relaxes this condition slightly: if the events are “mostly” independent and each event happens with “not too high” probability, then, with positive probability, none of the events will happen. Formally, let $B_1, B_2,…,B_n$ be events with $\Pr(B_i)\leqp$ that are mutually independent of all but d of others. If $p*e*(d+1)<1$, then $\Pr(\text{none of the events happens})>0$.

In this talk, a quantum (or rather linear algebraic) analogue of this lemma will be presented. Namely, let $X_1, X_2,…,X_n$ be subspaces such that each subspace is mutually “independent" of all but d of the others. If each space doesn’t “occupy” the whole space too much then the intersection of $X_i$’s is not empty. We will clarify and prove this statement in this talk.

This talk will be based on "A Quantum Lovasz Local Lemma” by Ambainis, Kempe, and Sattath (see
https://arxiv.org/pdf/0911.1696.pdf ).