SUMMARYIn this paper, we derive bounds on performance guarantees of online algorithms for real-time preemptive scheduling of jobs with deadlines on K machines when jobs are characterized in terms of their minimum stretch factor (or, equivalently, their maximum execution rate r = 1= ). We consider two well-known preemptive models that are of interest from practical applications: the hard real-time scheduling model in which a job must be completed if it was admitted for execution by the online scheduler, and the ÿrm real-time scheduling model in which the scheduler is allowed not to complete a job even if it was admitted for execution by the online scheduler. In both models, the objective is to maximize the sum of execution times of the jobs that were executed to completion, preemption is allowed, and the online scheduler must immediately decide, whenever a job arrives, whether to admit it for execution or reject it. However, migration of jobs is not allowed. We measure the competitive ratio of any online algorithm as the ratio of the value of the objective function obtained by this algorithm to that of the best possible o ine algorithm. We show that no online algorithm can have a competitive ratio greater than 1 − (1= ) + for 1 machine and 1 − (1=(K )) for K¿1 machines for hard real-time scheduling, and greater than 1 − (3=(4 )) + for ÿrm real-time scheduling on a single machine, where ¿0 may be arbitrarily small, even if the algorithm is allowed to know the value of in advance. On the other hand, we exhibit a simple online scheduler that achieves a competitive ratio of at least 1 − (1= ) in either of these models with K machines. The performance guarantee of our simple scheduler shows that it is in fact an optimal scheduler for hard real-time scheduling with 1 machine. We also describe an alternative scheduler for ÿrm real-time scheduling on a single machine in which the competitive ratio does not go to zero as approaches 1. Both of our schedulers do not know the value of in advance.

A 63-year-old female presented with a 12-week history of worsening proximal pain and stiffness. She was diagnosed with polymyalgia rheumatica and started on corticosteroids. The authors were unable to wean-off her steroid treatment, despite trying various steroid sparing agents on different occasions with no benefit. In August 2010, she was diagnosed with giant cell arteritis with a temporal artery biopsy and ultrasound of the temporal and axillary arteries. An fluorine-18-deoxyglucose positron emission tomography CT showed increased uptake in the aorta and major vessels, in keeping with widespread large vessel involvement. Due to the disease severity, the failure of previous disease-modifying agents and the development of steroid related sideeffects, the authors decided to treat her with intravenous tocilizumab (TCZ;an interleukin 6 blocker). After her first infusion, the patient reported excellent response with normalisation of her inflammatory markers. Prednisolone reduced from 20 mg to 3.5 mg /day after five infusions of TCZ (8 mg/kg).

δ-hyperbolic graphs, originally conceived by Gromov in 1987, occur often in many network applications; for fixed δ, such graphs are simply called hyperbolic graphs and include non-trivial interesting classes of "non-expander" graphs. The main motivation of this paper is to investigate the effect of the hyperbolicity measure δ on expansion and cut-size bounds on graphs (here δ need not be a constant), and the asymptotic ranges of δ for which these results may provide improved approximation algorithms for related combinatorial problems. To this effect, we provide constructive bounds on node expansions for δ-hyperbolic graphs as a function of δ, and show that many witnesses (subsets of nodes) for such expansions can be computed efficiently even if the witnesses are required to be nested or sufficiently distinct from each other. To the best of our knowledge, these are the first such constructive bounds proven. We also show how to find a large family of s-t cuts with relatively small number of cut-edges when s and t are sufficiently far apart. We then provide algorithmic consequences of these bounds and their related proof techniques for two problems for δ-hyperbolic graphs (where δ is a function f of the number of nodes, 2 Bhaskar DasGupta et al.the exact nature of growth of f being dependent on the particular problem considered). Mathematics Subject Classification1 IntroductionUseful insights for many complex systems such as the world-wide web, social networks, metabolic networks, and protein-protein interaction networks can often be obtained by representing them as parameterized networks and analyzing them using graph-theoretic tools. Some standard measures used for such investigations include degree based measures (e.g., maximum/minimum/average degree or degree distribution) connectivity based measures (e.g., clustering coefficient, claw-free property, largest cliques or densest sub-graphs), and geodesic based measures (e.g., diameter or betweenness centrality). It is a standard practice in theoretical computer science to investigate and categorize the computational complexities of combinatorial problems in terms of ranges of these parameters. For example:◮ Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. ◮ Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem.In this paper we consider a topological measure called Gromov-hyperbolicity (or, simply hyperbolicity for short) for undirected unweighted graphs that has recently received significant attention from researchers in both the graph theory and the network science community. This hyperbolicity measure δ was originally conceived in a somewhat different group-theoretic context by Gromov [20]. The measure was first defined for infinite continuous metric space via properties of geodesics [10], but was later also adopted for finite graphs. Lately, there have been...

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