Potential-Function Proofs for First-Order Methods

This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants

On the Lovász theta function for independent sets in sparse graphs

We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the Lovász ϑ-function based semidefinite program (SDP) has an integrality gap of O(d/log3/2 d), improving on the previous best result of O(d/log d). This improvement is based on a new Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show that for stronger SDPs, namely, those obtained using polylog(d) levels of the SA+ semidefinite hierarchy, the integrality gap reduces to O(d/log2 d). This matches the best unique-games-based hardness result up to lower-order poly(log log d) factors. Finally, we give an algorithmic version of this SA+-based integrality gap result, albeit using d levels of SA+, via a coloring algorithm of Johansson

Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box [0,L]3 is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t)=(ϖ−10Ωm)αm for 1≤m≤∞ where αm=2m/(4m−3) and [Ωm(t)]2m=L−3∫V|ω|2mdV with ϖ0=νL−2. All four simulations unexpectedly show that the Dm are ordered for m=1,…,9 such that Dm+1<Dm. Moreover, the Dm squeeze together such that Dm+1/Dm↗1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 40963

Two-dimensional, homogeneous, isotropic fluid turbulence with polymer additives

We present the most extensive direct numerical simulations, attempted so far, of statistically steady, homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental studies; and we propose new experiments.Comment: 8 pages, 8 figure

Secretary Problems: Weights and Discounts

The classical secretary problem studies the problem of selecting online an element (a “secretary”) with maximum value in a randomly ordered sequence. The difficulty lies in the fact that an element must be either selected or discarded upon its arrival, and this decision is irrevocable. Constant-competitive algorithms are known for the classical secretary problems and several variants. We study the following two extensions of the secretary problem: In the discounted secretary problem, there is a time-dependent “discount” factor $d(t)$, and the benefit derived from selecting an element/secretary e at time t is $d(t)v(e)$. For this problem with arbitrary (not necessarily decreasing) functions $d(t)$, we show a constant-competitive algorithm when the expected optimum is known in advance. With no prior knowledge, we exhibit a lower bound and give a nearly matching $O(log n)$-competitive algorithm. In the weighted secretary problem, up to K secretaries can be selected; when a secretary is selected (s)he must be irrevocably assigned to one of K positions, with position k having weight $w(k)$, and assigning object/secretary e to position k has benefit $w(k)v(e)$. The goal is to select secretaries and assign them to positions to maximize the sum of $w(k)v(e_k)$ where $e_k$ is the secretary assigned to position k. We give constant-competitive algorithms for this problem. Most of these results can also be extended to the matroid secretary case for a large family of matroids with a constant-factor loss, and an O(log rank) loss for general matroids. These results are based on a reduction from various matroids to partition matroids which present a unified approach to many of the upper bounds of Babaioff et al. These problems have connections to online mechanism design. All our algorithms are monotone, and hence lead to truthful mechanisms for the corresponding online auction problems

Interpreting COVID-19 Deaths among Nursing Home Residents in the US: The Changing Role of Facility Quality over Time

A report published last year by the Centers for Medicare & Medicaid Services (CMS) highlighted that COVID-19 case counts are more likely to be high in lower quality nursing homes than in higher quality ones. Since then, multiple studies have examined this association with a handful also exploring the role of facility quality in explaining resident deaths from the virus. Despite this wide interest, no previous study has investigated how the relation between quality and COVID-19 mortality among nursing home residents may have changed, if at all, over the progression of the pandemic. This understanding is indeed lacking given that prior studies are either cross-sectional or are analyses limited to one specific state or region of the country. To address this gap, we analyzed changes in nursing home resident deaths across the US between June 1, 2020 and January 31, 2021 (n = 12,415 nursing homes X 8 months) using both descriptive and multivariable statistics. We merged publicly available data from multiple federal agencies with mortality rate (per 100,000 residents) as the outcome and CMS 5-star quality rating as the primary explanatory variable of interest. Covariates, based on the prior literature, consisted of both facility- and community-level characteristics. Findings from our secondary analysis provide robust evidence of the association between nursing home quality and resident deaths due to the virus diminishing over time. In connection, we discuss plausible reasons, especially duration of staff shortages, that over time might have played a critical role in driving the quality-mortality convergence across nursing homes in the US

Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin