The Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

Approximation Algorithms for Flexible Graph Connectivity

We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), and IPCO 2020: pp. 13-26). Let $k\geq 1$, $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges $S$ and a set of unsafe edges $U$, and nonnegative costs $c: E\to\Re$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any subset $F'$ of unsafe edges with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible solution $F$ that minimizes $c(F) = \sum_{e \in F} c_e$. We present a simple $2$-approximation algorithm for the $(1,1)$-FGC problem via a reduction to the minimum-cost rooted $2$-arborescence problem. This improves on the $2.527$-approximation algorithm of Adjiashvili et al. Our $2$-approximation algorithm for the $(1,1)$-FGC problem extends to a $(k+1)$-approximation algorithm for the $(1,k)$-FGC problem. We present a $4$-approximation algorithm for the $(p,1)$-FGC problem, and an $O(q\log|V|)$-approximation algorithm for the $(p,q)$-FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted $(1,1)$-FGC problem by presenting a $16/11$-approximation algorithm. The $(p,q)$-FGC problem is related to the well-known Capacitated $k$-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of Capacitated Network Design. We give a $\min(k,2 u_{max})$-approximation algorithm for the Cap-k-ECSS problem, where $u_{max}$ denotes the maximum capacity of an edge.Comment: 23 pages, 1 figure, preliminary version in the Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), December 15-17, (LIPIcs, Volume 213, Article No. 9, pp. 9:1-9:14), see https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9. Related manuscript: arXiv:2102.0330

A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case

Given a connected undirected graph $\bar{G}$ on $n$ vertices, and non-negative edge costs $c$, the 2ECM problem is that of finding a $2$-edge~connected spanning multisubgraph of $\bar{G}$ of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of $\bar{G}$, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution $x$, Carr and Ravi (1998) showed that the integrality gap is at most $\frac43$: they show that the vector $\frac43 x$ dominates a convex combination of incidence vectors of $2$-edge connected spanning multisubgraphs of $\bar{G}$. We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lov\'{a}sz's splitting-off theorem. Our proof naturally leads to a $\frac43$-approximation algorithm for half-integral instances. Given a half-integral solution $x$ to the LP for 2ECM, we give an $O(n^2)$-time algorithm to obtain a $2$-edge connected spanning multisubgraph of $\bar{G}$ whose cost is at most $\frac43 c^T x$

Chromatin Profiles of Chromosomally Integrated Human Herpesvirus-6A

Human herpesvirus-6A (HHV-6A) and 6B (HHV-6B) are two closely related betaherpesviruses that are associated with various diseases including seizures and encephalitis. The HHV-6A/B genomes have been shown to be present in an integrated state in the telomeres of latently infected cells. In addition, integration of HHV-6A/B in germ cells has resulted in individuals harboring this inherited chromosomally integrated HHV-6A/B (iciHHV-6) in every cell of their body. Until now, the viral transcriptome and the epigenetic modifications that contribute to the silencing of the integrated virus genome remain elusive. In the current study, we used a patient-derived iciHHV-6A cell line to assess the global viral gene expression profile by RNA-seq, and the chromatin profiles by MNase-seq and ChIP-seq analyses. In addition, we investigated an in vitro generated cell line (293-HHV-6A) that expresses GFP upon the addition of agents commonly used to induce herpesvirus reactivation such as TPA. No viral gene expression including miRNAs was detected from the HHV-6A genomes, indicating that the integrated virus is transcriptionally silent. Intriguingly, upon stimulation of the 293-HHV-6A cell line with TPA, only foreign promoters in the virus genome were activated, while all HHV-6A promoters remained completely silenced. The transcriptional silencing of latent HHV-6A was further supported by MNase-seq results, which demonstrate that the latent viral genome resides in a highly condensed nucleosome-associated state. We further explored the enrichment profiles of histone modifications via ChIP-seq analysis. Our results indicated that the HHV-6 genome is modestly enriched with the repressive histone marks H3K9me3/H3K27me3 and does not possess the active histone modifications H3K27ac/H3K4me3. Overall, these results indicate that HHV-6 genomes reside in a condensed chromatin state, providing insight into the epigenetic mechanisms associated with the silencing of the integrated HHV-6A genome

Microbial control of diatom bloom dynamics in the open ocean

Diatom blooms play a central role in supporting foodwebs and sequestering biogenic carbon to depth. Oceanic conditions set bloom initiation, whereas both environmental and ecological factors determine bloom magnitude and longevity. Our study reveals another fundamental determinant of bloom dynamics. A diatom spring bloom in offshore New Zealand waters was likely terminated by iron limitation, even though diatoms consumed <1/3 of the mixed-layer dissolved iron inventory. Thus, bloom duration and magnitude were primarily set by competition for dissolved iron between microbes and small phytoplankton versus diatoms. Significantly, such a microbial mode of control probably relies both upon out-competing diatoms for iron (i.e., K-strategy), and having high iron requirements (i.e., r-strategy). Such resource competition for iron has implications for carbon biogeochemistry, as, blooming diatoms fixed three-fold more carbon per unit iron than resident non-blooming microbes. Microbial sequestration of iron has major ramifications for determining the biogeochemical imprint of oceanic diatom blooms. Citation: Boyd, P. W., et al. (2012), Microbial control of diatom bloom dynamics in the open ocean, Geophys. Res. Lett., 39, L18601

Socio-economic factors associated with loss to follow-up among individuals with HCV:A Dutch nationwide cross-sectional study

Background and Aims: The path to hepatitis C virus (HCV) elimination is complicated by individuals who become lost to follow-up (LTFU) during care, particularly before receiving effective HCV treatment. We aimed to determine factors contributing to LTFU and whether LTFU is associated with mortality. Methods: In this secondary analysis, we constructed a database including individuals with HCV who were either LTFU (data from the nationwide HCV retrieval project, CELINE) or treated with directly acting antivirals (DAA) (data from Statistics Netherlands) between 2012 and 2019. This database was linked to mortality data from Statistics Netherlands. Determinants associated with being LTFU versus DAA-treated were assessed using logistic regression, and mortality rates were compared between groups using exponential survival models. These analyses were additionally stratified on calendar periods: 2012–2014, 2015–2017 and 2018–2019. Results: About 254 individuals, LTFU and 5547 DAA-treated were included. Being institutionalized (OR = 5.02, 95% confidence interval (CI) = 3.29–7.65), household income below the social minimum (OR = 1.96, 95% CI = 1.25–3.06), receiving benefits (OR = 1.74, 95% CI = 1.20–2.52) and psychiatric comorbidity (OR = 1.51, 95% CI = 1.09–2.10) were associated with LTFU. Mortality rates were significantly higher in individuals LTFU compared to those DAA-treated (2.99 vs. 1.15/100 person-years (PY), p &lt;.0001), while in those DAA-treated, mortality rates slowly increased between 2012–2014 (.22/100PY) and 2018–2019 (2.25/100PY). Conclusion: In the Netherlands, individuals who are incarcerated/institutionalized, with low household income, or with psychiatric comorbidities are prone to being LTFU, which is associated with higher mortality. HCV care needs to be adapted for these vulnerable individuals.</p

Prospectus, December 6, 1974

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