We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of $w$ workers and a multiset $T \subseteq[t]$ of $|T|=w$ tasks, a memoryless worker-task assignment function is any function $\phi$ that assigns the workers $[w]$ to the tasks $T$ based only on the current value of $T$. The assignment function $\phi$ is said to have switching cost at most $k$ if, for every task multiset $T$, changing the contents of $T$ by one task changes $\phi(T)$ by at most $k$ worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost $O(\log w \log (wt))$ and an explicit construction that achieves switching cost $\operatorname{polylog} (wt)$. We also prove a super-constant lower bound on switching cost: we show that for any value of $w$, there exists a value of $t$ for which the optimal switching cost is $w$. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of $w$. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.
翻译:我们研究将无记忆工人分配到需求动态变化的任务中的基本问题。 鉴于一组以美元计的工人和以美元计的多设的美元(T),无记忆工人-任务分配功能是将工人[W]美元分配到任务中的任何函数$(Phip$),它仅仅根据目前美元的价值来计算。 分配函数$(phi$)据说以最多美元转换成本,如果每个任务多设的美元(T),用一个任务改变$(T),用美元计的美元改变美元。无记忆工人-任务分配的目的是用最小的切换成本来构建一个任务函数。在过去的工作中,确定最佳转换成本的问题一直是一个尚未解决的问题。没有已知的亚线性上值问题,经过相当大的努力,已知的低约束仍然是4美元(CentrialP 2020)。我们表明,实现多功能转换成本成本(美元)是有可能的。我们通过一个固定的转换成本(美元),我们通过一个固定的方法进行一个固定的转换一个成本(美元)的构建一个成本(美元),它可以实现一个真正的变现成本。