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We present two results on slime mold computations. In wet-lab experiments (Nature00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems. Biologists proposed a mathematical model, a system of differential equations, for the slimes adaption process (J. Theoretical Biology07). It was shown that the process convergences to the shortest path (J. Theoretical Biology12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector. Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can $varepsilon$-approximately solve linear programs with positive cost vector (ITCS16). Their analysis requires a feasible starting point, a step size depending linearly on $varepsilon$, and a number of steps with quartic dependence on $mathrm{opt}/(varepsilonPhi)$, where $Phi$ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost ($mathrm{opt}$). We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of $varepsilon$, and the number of steps depends logarithmically on $1/varepsilon$ and quadratically on $mathrm{opt}/Phi$.
We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protoc
Hagfish slime is a unique predator defense material containing a network of long fibrous threads each ~ 10 cm in length. Hagfish release the threads in a condensed coiled state known as thread cells, or skeins (~ 100 microns), which must unravel with
We present a number of new results about range searching for colored (or categorical) data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(nmathop{rm polylog}n)$ space that can report the dist
We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. We make use of the rich symmetry structure to develop a novel set of tools for
In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $mathcal{M}={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is