In the Online Machine Covering problem jobs, defined by their sizes, arrive one by one and have to be assigned to $m$ parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. In this work, we study the Machine Covering problem in the recently popular random-order model. Here no extra resources are present, but instead the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham's Greedy-strategy in this context and establish that its competitive ratio decreases slightly to $\Theta\left(\frac{m}{\log(m)}\right)$ which is asymptotically tight. Then, as our main result, we present an improved $\tilde{O}(\sqrt[4]{m})$-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound showing that no algorithm can have a competitive ratio of $O\left(\frac{\log(m)}{\log\log(m)}\right)$ in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest.
翻译:在线机器 覆盖有问题的工作, 按其大小定义, 一个一个一个接一个, 并且必须分配到 $m 的平行和相同的机器, 目标是最大限度地增加最不载荷的机器的负荷。 在此工作中, 我们研究最近流行的随机命令模式中的机器覆盖问题。 这里没有额外的资源, 但对手被削弱, 因为它只能决定输入数据集, 而工作以随机方式统一披露。 它对于机器覆盖下限通常与高度结构化输入序列相关。 我们首先分析格雷厄姆在此背景下的贪婪战略, 并确定其竞争性比率略微降低到$\ Theta\left (\ frac{m\\ log\\\\\ r\\\\ right), 因为它过于紧凑紧。 然后, 作为我们的主要结果, 我们只能提出一个改进的 $tilde{O} (sqrt[ 4{m} $- 竞争性算法。 这个问题的结果是通过利用来自随机的工作顺序的额外信息实现的。 使用取样技术, 来设定一个更低的运算法的模型, 我们无法对一个较高级的运算法进行相对的系统进行区分。