Finding Almost Tight Witness Trees

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs

Choose your witnesses wisely

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs.Comment: 33 pages, 7 figures, submitted to IPCO 202

Approximations for Throughput Maximization

In this paper we study the classical problem of throughput maximization. In this problem we have a collection $J$ of $n$ jobs, each having a release time $r_j$, deadline $d_j$, and processing time $p_j$. They have to be scheduled non-preemptively on $m$ identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their $[r_j,d_j]$ window. This problem has been studied extensively (even for the case of $m=1$). Several special cases of the problem remain open. Bar-Noy et al. [STOC1999] presented an algorithm with ratio $1-1/(1+1/m)^m$ for $m$ machines, which approaches $1-1/e$ as $m$ increases. For $m=1$, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio $1-\frac{1}{e}-\varepsilon$ (for any $\varepsilon>0$). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio $1-1/e-\varepsilon_0$ for some absolute constant $\varepsilon_0>0$ for any fixed $m$. They also presented an algorithm with ratio $1-O(\sqrt{\log m/m})-\varepsilon$ for general $m$ which approaches 1 as $m$ grows. The approximability of the problem for $m=O(1)$ remains a major open question. Even for the case of $m=1$ and $c=O(1)$ distinct processing times the problem is open (Sgall [ESA2012]). In this paper we study the case of $m=O(1)$ and show that if there are $c$ distinct processing times, i.e. $p_j$'s come from a set of size $c$, then there is a $(1-\varepsilon)$-approximation that runs in time $O(n^{mc^7\varepsilon^{-6}}\log T)$, where $T$ is the largest deadline. Therefore, for constant $m$ and constant $c$ this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning

The intersection of video capsule endoscopy and artificial intelligence: addressing unique challenges using machine learning

Introduction: Technical burdens and time-intensive review processes limit the practical utility of video capsule endoscopy (VCE). Artificial intelligence (AI) is poised to address these limitations, but the intersection of AI and VCE reveals challenges that must first be overcome. We identified five challenges to address. Challenge #1: VCE data are stochastic and contains significant artifact. Challenge #2: VCE interpretation is cost-intensive. Challenge #3: VCE data are inherently imbalanced. Challenge #4: Existing VCE AIMLT are computationally cumbersome. Challenge #5: Clinicians are hesitant to accept AIMLT that cannot explain their process. Methods: An anatomic landmark detection model was used to test the application of convolutional neural networks (CNNs) to the task of classifying VCE data. We also created a tool that assists in expert annotation of VCE data. We then created more elaborate models using different approaches including a multi-frame approach, a CNN based on graph representation, and a few-shot approach based on meta-learning. Results: When used on full-length VCE footage, CNNs accurately identified anatomic landmarks (99.1%), with gradient weighted-class activation mapping showing the parts of each frame that the CNN used to make its decision. The graph CNN with weakly supervised learning (accuracy 89.9%, sensitivity of 91.1%), the few-shot model (accuracy 90.8%, precision 91.4%, sensitivity 90.9%), and the multi-frame model (accuracy 97.5%, precision 91.5%, sensitivity 94.8%) performed well. Discussion: Each of these five challenges is addressed, in part, by one of our AI-based models. Our goal of producing high performance using lightweight models that aim to improve clinician confidence was achieved

Node connectivity augmentation via iterative randomized rounding

Many network design problems deal with the design of low-cost networks that are resilient to the failure of their elements (such as nodes or links). One such problem is Connectivity Augmentation, with the goal of cheaply increasing the (edge- or node-)connectivity of a given network from a value k to k+ 1. The problem is NP-hard for k≥ 1 , and the most studied setting focuses on the case of edge-connectivity with k= 1. In this work, we give a 1.892-approximation algorithm for the NP-hard problem of augmenting the node-connectivity of any given graph from 1 to 2, which improves upon the state-of-the-art approximation previously developed in the literature. The starting point of our work is a known reduction from Connectivity Augmentation to some specific instances of the Node-Steiner Tree problem, and our result is obtained by developing a new and simple analysis of the iterative randomized rounding technique when applied to such Steiner Tree instances. Our results also imply a 1.892-approximation algorithm for the problem of augmenting the edge-connectivity of a given graph from any value k to k+ 1. While this does not beat the best approximation factor known for this problem, a key point of our work is that the analysis of our approximation factor is less involved when compared to previous results in the literature. In addition, our work gives new insights on the iterative randomized rounding method, that might be of independent interest

Kinetic Study of Living Ring-Opening Metathesis Polymerization with Third-Generation Grubbs Catalysts

The rate of living ring-opening metathesis polymerization (ROMP) of <i>N</i>-hexyl-<i>exo</i>-norbornene-5,6-dicarboximide initiated by Grubbs third-generation catalyst precursors [(H₂IMes)­(py)₂(Cl)₂RuCHPh] and [(H₂IMes)­(3-Br-py)₂(Cl)₂RuCHPh] is measured to be independent of catalyst concentration. This result led to the development of a rate law describing living ROMP initiated by a Grubbs third-generation catalyst that includes an inverse first-order dependency in pyridine. Additionally, it is demonstrated that one of the two pyridines coordinated to the solid catalyst is fully dissociated in solution. The monopyridine adduct formation is confirmed in solution by ¹H DOSY (diffusion-ordered NMR spectroscopy), and a Van’t Hoff analysis of the equilibrium between mono- and dipyridine adducts (extrapolated <i>K</i>_eq,0 ∼ 0.5 at 25 °C). Finally, the difference in polymerization rates between two catalyst precursors is demonstrated to correspond to the difference in coordination strength between the two pyridines, suggesting that the catalytic species involved in the polymerization’s rate-determining step is not coordinated to pyridine

Kinetic Study of Living Ring-Opening Metathesis Polymerization with Third-Generation Grubbs Catalysts

The rate of living ring-opening metathesis polymerization (ROMP) of <i>N</i>-hexyl-<i>exo</i>-norbornene-5,6-dicarboximide initiated by Grubbs third-generation catalyst precursors [(H₂IMes)­(py)₂(Cl)₂RuCHPh] and [(H₂IMes)­(3-Br-py)₂(Cl)₂RuCHPh] is measured to be independent of catalyst concentration. This result led to the development of a rate law describing living ROMP initiated by a Grubbs third-generation catalyst that includes an inverse first-order dependency in pyridine. Additionally, it is demonstrated that one of the two pyridines coordinated to the solid catalyst is fully dissociated in solution. The monopyridine adduct formation is confirmed in solution by ¹H DOSY (diffusion-ordered NMR spectroscopy), and a Van’t Hoff analysis of the equilibrium between mono- and dipyridine adducts (extrapolated <i>K</i>_eq,0 ∼ 0.5 at 25 °C). Finally, the difference in polymerization rates between two catalyst precursors is demonstrated to correspond to the difference in coordination strength between the two pyridines, suggesting that the catalytic species involved in the polymerization’s rate-determining step is not coordinated to pyridine