Alien Registration- Carlson, Charlie (Strong, Franklin County)

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Improved Distributed Algorithms for Random Colorings

Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph $G$ of maximum degree $\Delta$ and an integer $k\geq\Delta+1$, the goal is to generate a random proper vertex $k$-coloring of $G$. Jerrum (1995) proved that the Glauber dynamics has $O(n\log{n})$ mixing time when $k>2\Delta$. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in $O(\log{n})$ rounds for $k>(2+\varepsilon)\Delta$ for any $\varepsilon>0$. We improve this result to $k>(11/6-\delta)\Delta$ for a fixed $\delta>0$. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.Comment: 25 pages, 2 figure

Approximate Counting and Expansion

Spin models are mathematical frameworks used to study the collective behavior of interacting particles in various systems. A model assigns a discrete spin to each vertex of an underlying graph and assigns a value to each spin assignment. This produces a natural counting problem, summing the value of all spin assignments. In this thesis we study the approximate counting problems of two specific spin models. First, we consider a variant of the ferromagnetic Ising model. The approximate counting problem of the ferromagnetic Ising model is tractable on all graphs (Jerrum and Sinclair 1993) but here we show that hidden inside this model are hard computational problems. Namely, we present computational thresholds for the approximate counting problem at fixed magnetization (that is, fixing the number of specific spins in each assignment). Next, we consider the ferromagnetic Potts model on graphs with expansion grantees. We give algorithms for the approximate counting problems on d-regular expanding graphs and small-set expander graphs. The approximate counting problem on bounded-degree graphs is as hard as counting independent sets in bipartite graphs (#BIS-hard), so our algorithms can be seen as evidence that hard instances of #BIS are rare.</p

A quantum advantage over classical for local max cut

We compare the performance of a quantum local algorithm to a similar classical counterpart on a well-established combinatorial optimization problem LocalMaxCut. We show that a popular quantum algorithm first discovered by Farhi, Goldstone, and Gutmannn [1] called the quantum optimization approximation algorithm (QAOA) has a computational advantage over comparable local classical techniques on degree-3 graphs. These results hint that even small-scale quantum computation, which is relevant to the current state-of the art quantum hardware, could have significant advantages over comparably simple classical computation

Diode laser array

A diode laser array comprises a substrate of a semiconductor material having first and second opposed surfaces. On the first surface is a plurality of spaced gain sections and a separate distributed Bragg reflector passive waveguide at each end of each gain section and optically connecting the gain sections. Each gain section includes a cavity therein wherein charge carriers are generated and recombine to generate light which is confined in the cavity. Also, the cavity, which is preferably a quantum well cavity, provides both a high differential gain and potentially large depth of loss modulation. Each waveguide has a wavelength which is preferably formed by an extension of the cavity of the gain sections and a grating. The grating has a period which provides a selective feedback of light into the gain sections to supporting lasing, which allows some of the light to be emitted from the waveguide normal to the surface of the substrate and which allows optical coupling of the gain sections. Also, the grating period provides an operating wavelength which is on the short wavelength side of the gain period of the gain sections required for laser oscillation. An RF pulse is applied so as to maximize the magnitude of the loss modulation and the differential gain in the gain sections. The array is operated by applying a DC bias to all the gain sections at a level just below the threshold of the gain sections to only one of the gain sections which raises the bias in all of the gain sections to a level that causes all of the gain sections to oscillate. Thus, a small bias can turn the array on and off

Approximation Algorithms for Norm Multiway Cut

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway, a generalization of the problem, in which the goal is to minimize the $\ell_p$ norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2} n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$ approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2} k)$ approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random Conjecture.Comment: 25 pages, ESA 202

Approximation Algorithm for Norm Multiway Cut

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced ?_p-norm Multiway Cut, a generalization of the problem, in which the goal is to minimize the ?_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} nlog^{1/2+1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4-?} approximation algorithm for every ? > 0 assuming the Hypergraph Dense-vs-Random Conjecture

Approximation Algorithms for Quantum Max-dd-Cut

We initiate the algorithmic study of the Quantum Max-$d$-Cut problem, a quantum generalization of the well-known Max-$d$-Cut problem. The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. Equivalently, this problem is physically motivated by the $SU(d)$-Heisenberg model, a spin glass model that generalized the well-known Heisenberg model over qudits. We develop a polynomial-time randomized approximation algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees. Moreover, we prove the tightness of our analysis by presenting an algorithmic gap instance for Quantum Max-d-Cut problem with $d \geq 3$.Comment: 42 page

Algorithms for the ferromagnetic Potts model on expanders

We give algorithms for approximating the partition function of the ferromagnetic Potts model on $d$-regular expanding graphs. We require much weaker expansion than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models, using extremal graph theory and applications of Karger's algorithm to counting cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We believe that these methods can shed more light on other important problems such as sub-exponential algorithms for approximate counting problems.Comment: 27 page

The Effective Use of Sealing Technology to Meet the Ever-Tightening Environmental Regulations

Short Cours