A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph G with n vertices so that, for any edge set F of size |F| ≤ f, one can approximate the distance between p and q in G ∖ F by reading only the labels of F ∪ {p,q}.

We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow.

This paper addresses point-to-point packet routing in undirected networks, which is the most important communication primitive in most networks. The main result proves the existence of routing tables that deterministically guarantee a polylog-competitive completion-time:

The packet routing problem asks to select routing paths that minimize the maximum edge congestion for a set of packets specified by source-destination vertex pairs. We revisit a semi-oblivious approach to this problem: each source-destination pair is assigned a small set of well-chosen predefined paths before the demand is revealed, while the sending rates along the paths can be optimally adapted to the demand. This approach has been considered in practice in network traffic engineering due to its superior robustness and performance as compared to both oblivious routing and traditional traffic engineering approaches.

We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the permanent upper bound with a linear combination of the subproblem bounds at a moderately large depth of the recursion tree. This method, we call deep rejection sampling, is empirically shown to outperform the basic, depth-zero variant, as well as a related method by Kuck et al. (NeurIPS 2019). We analyze the expected running time of the scheme on random $(0, 1)$-matrices where each entry is independently $1$ with probability $p$. Our bound is superior to a previous one for $p$ less than $1/5$, matching another bound that was only known to hold when every row and column has density exactly $p$.