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

Mycobacterium indicus pranii Supernatant Induces Apoptotic Cell Death in Mouse Peritoneal Macrophages In Vitro

Mycobacterium indicus pranii (MIP), also known as Mw, is a saprophytic, non-pathogenic strain of Mycobacterium and is commercially available as a heat-killed vaccine for leprosy and recently tuberculosis (TB) as part of MDT. In this study we provide evidence that cell-free supernatant collected from original MIP suspension induces rapid and enhanced apoptosis in mouse peritoneal macrophages in vitro. It is demonstrated that the MIP cell-free supernatant induced apoptosis is mitochondria-mediated and caspase independent and involves mitochondrial translocation of Bax and subsequent release of AIF and cytochrome c from the mitochondria. Experiments with pharmacological inhibitors suggest a possible role of PKC in mitochondria-mediated apoptosis of macrophages

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Balancing vectors in any norm

In the vector balancing problem, we are given symmetric convex bodies C and K in ℝn, and our goal is to determine the minimum number β ≥ 0, known as the vector balancing constant from C to K, such that for any sequence of vectors in C there always exists a signed combination of them lying inside βK. Many fundamental results in discrepancy theory, such as the Beck-Fiala theorem (Discrete Appl. Math '81), Spencer's "six standard deviations suffice" theorem (Trans. Amer. Math. Soc '85) and Banaszczyk's vector balancing theorem (Random Structures & Algorithms '98) correspond to bounds on vector balancing constants. The above theorems have inspired much research in recent years within theoretical computer science. In this work, we show that all vector balancing constants admit "good" approximate characterizations, with approximation factors depending only polylogarithmically on the dimension n. First, we show that a volumetric lower bound due to Banaszczyk is tight within a O(log n) factor. Our proof is algorithmic, and we show that Rothvoss's (FOCS '14) partial coloring algorithm can be analyzed to obtain these guarantees. Second, we present a novel convex program which encodes the "best possible way" to apply Banaszczyk's vector balancing theorem for bounding vector balancing constants from above, and show that it is tight within an O(log2.5 n) factor. This also directly yields a corresponding polynomial time approximation algorithm both for vector balancing constants, and for the hereditary discrepancy of any sequence of vectors with respect to an arbitrary norm