Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{FS_2P}$ algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result, we obtain the circuit lower bound $\mathsf{S_2E} \not\subset \mathsf{SIZE}[2^n/n]$ and many other corollaries: 1. Near-maximum circuit lower bound for $\mathsf{\Sigma_2E} \cap \mathsf{\Pi_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$. 2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings

The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-\epsilon)$-approximate solution in $O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+\epsilon)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}{\epsilon}\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm \epsilon)$-approximate solutions require $\Omega\left(\frac{\log n}{\epsilon}\right)$ rounds to compute.Comment: To appear in PODC 202

Broadband negative refraction in stacked fishnet metamaterial

We demonstrate a scheme to utilize the stacked fishnet metamaterial for all-angle negative refraction and subwavelength imaging within a wide frequency range starting from zero frequency. The theoretical predictions are verified by the finite-difference-in-time-domain (FDTD) numerical simulations. The phenomena come from the negative evanescent coupling between the adjacent slab waveguides through the breathing air holes perforated on metal layers.Comment: 8 pages, 4 figure