In this paper, a single-machine semi-online scheduling problem with rejection is addressed. In this model, “semi-online” implies that [Formula: see text], where [Formula: see text] and [Formula: see text] are the maximum processing time and the minimum one among all jobs, respectively, [Formula: see text]. In this setting, each job arrives online over time, and rejection is allowable. Our goal is minimizing the total penalty cost of rejected jobs plus the total completion time of processed jobs. The seminal result of this study is that we offer an algorithm with competitive ratio [Formula: see text], which matches the result of the problem without rejection.
In this paper, we discuss several cost-sharing methods for cover-set games, for example, approximately budget-balanced, in the core, and/or strategyproof. Different from the traditional set cover game, in cover-set game, the elements stand for services, and each agent is both a service provider and receiver. In this study, for unselfish agents, we design a cost allocation method that can recover no less than [Formula: see text] fraction of the total cost, where [Formula: see text] and [Formula: see text] is the maximum number of receiving services among all agents. In addition, for agents who are selfish service providers with privately known valuations, we present a strategyproof charging mechanism, further, the total cost is no more than [Formula: see text] times that of an optimal solution.
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