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This paper considers elections in which voters choose one candidate each, independently according to known probability distributions. A candidate receiving a strict majority (absolute or relative, depending on the version) wins. After the voters have made their choices, each vote can be inspected to determine which candidate received that vote. The time (or cost) to inspect each of the votes is known in advance. The task is to (possibly adaptively) determine the order in which to inspect the votes, so as to minimize the expected time to determine which candidate has won the election. We design polynomial-time constant-factor approximation algorithms for both the absolute-majority and the relative-majority version. Both algorithms are based on a two-phase approach. In the first phase, the algorithms reduce the number of relevant candidates to O(1), and in the second phase they utilize techniques from the literature on stochastic function evaluation to handle the remaining candidates. In the case of absolute majority, we show that the same can be achieved with only two rounds of adaptivity. 

Abstract We study a fundamental online job admission problem where jobs with deadlines arrive online over time at their release dates, and the task is to determine a preemptive single-server schedule which maximizes the number of jobs that complete on time. To circumvent known impossibility results, we make a standard slackness assumption by which the feasible time window for scheduling a job is at least $$1+\varepsilon $$ 1 + ε times its processing time, for some $$\varepsilon >0$$ ε > 0 . We quantify the impact that different provider commitment requirements have on the performance of online algorithms. Our main contribution is one universal algorithmic framework for online job admission both with and without commitments. Without commitment, our algorithm with a competitive ratio of  $$\mathcal {O}(1/\varepsilon )$$ O ( 1 / ε ) is the best possible (deterministic) for this problem. For commitment models, we give the first non-trivial performance bounds. If the commitment decisions must be made before a job’s slack becomes less than a $$\delta $$ δ -fraction of its size, we prove a competitive ratio of $$\mathcal {O}(\varepsilon /((\varepsilon -\delta )\delta ^2))$$ O ( ε / ( ( ε - δ ) δ 2 ) ) , for $$0<\delta <\varepsilon $$ 0 < δ < ε . When a provider must commit upon starting a job, our bound is  $$\mathcal {O}(1/\varepsilon ^2)$$ O ( 1 / ε 2 ) . Finally, we observe that for scheduling with commitment the restriction to the “unweighted” throughput model is essential; if jobs have individual weights, we rule out competitive deterministic algorithms.