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Register a new userThe Best-or-Worst and the Postdoc problems
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We consider two variants of the secretary problem, theemph{ Best-or-Worst} and the emph{Postdoc} problems, which are closely related. First, we prove that both variants, in their standard form with binary payoff 1 or 0, share the same optimal stopping rule. We also consider additional cost/perquisites depending on the number of interviewed candidates. In these situations the optimal strategies are very different. Finally, we also focus on the Best-or-Worst variant with different payments depending on whether the selected candidate is the best or the worst.
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In this paper we consider two variants of the Secretary problem: The Best-or-Worst and the Postdoc problems. We extend previous work by considering that the number of objects is not known and follows either a discrete Uniform distribution $mathcal{U}[1,n]$ or a Poisson distribution $mathcal{P}(lambda)$. We show that in any case the optimal strategy is a threshold strategy, we provide the optimal cutoff values and the asymptotic probabilities of success. We also put our results in relation with closely related work.
We compute the best constant in the Khintchine inequality under assumption that the sum of Rademacher random variables is zero.
Flow routing over inter-datacenter networks is a well-known problem where the network assigns a path to a newly arriving flow potentially according to the network conditions and the properties of the new flow. An essential system-wide performance metric for a routing algorithm is the flow completion times, which affect the performance of applications running across multiple datacenters. Current static and dynamic routing approaches do not take advantage of flow size information in routing, which is practical in a controlled environment such as inter-datacenter networks that are managed by the datacenter operators. In this paper, we discuss Best Worst-case Routing (BWR), which aims at optimizing the tail completion times of long-running flows over inter-datacenter networks with non-uniform link capacities. Since finding the path with the best worst-case completion time for a new flow is NP-Hard, we investigate two heuristics, BWRH and BWRHF, which use two different upper bounds on the worst-case completion times for routing. We evaluate BWRH and BWRHF against several real WAN topologies and multiple traffic patterns. Although BWRH better models the BWR problem, BWRH and BWRHF show negligible difference across various system-wide performance metrics, while BWRHF being significantly faster. Furthermore, we show that compared to other popular routing heuristics, BWRHF can reduce the mean and tail flow completion times by over $1.5times$ and $2times$, respectively.
The reader is reminded of several puzzles involving randomness. These may be ill-posed, and if well-posed there is sometimes a solution that uses probabilistic intuition in a special way. Various examples are presented including the well known problem of the lost boarding pass: what is the probability that the last passenger boarding a fully booked plane sits in the assigned seat if the first passenger has occupied a randomly chosen seat? This problem, and its striking answer of $frac12$, has attracted a good deal of attention since around 2000. We review elementary solutions to this, and to the more general problem of finding the probability the $m$th passenger sits in the assigned seat when in the presence of some number $k$ of passengers with lost boarding passes. A simple proof is presented of the independence of the occupancy status of different seats, and a connection to the Poisson--Dirichlet distribution is mentioned.
Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.
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