We study the problem of distributed task allocation in multi-agent systems. Suppose there is a collection of agents, a collection of tasks, and a demand vector, which specifies the number of agents required to perform each task. The goal of the agents is to cooperatively allocate themselves to the tasks to satisfy the demand vector. We study the dynamic version of the problem where the demand vector changes over time. Here, the goal is to minimize the switching cost, which is the number of agents that change tasks in response to a change in the demand vector. The switching cost is an important metric since changing tasks may incur significant overhead. We study a mathematical formalization of the above problem introduced by Su, Su, Dornhaus, and Lynch, which can be reformulated as a question of finding a low distortion embedding from symmetric difference to Hamming distance. In this model it is trivial to prove that the switching cost is at least 2. We present the first non-trivial lower bounds for the switching cost, by giving lower bounds of 3 and 4 for different ranges of the parameters.

中文翻译：

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动态分布式任务分配的下限

我们研究了多代理系统中的分布式任务分配问题。假设有一组代理、一组任务和一个需求向量，它指定执行每个任务所需的代理数量。代理的目标是将自己协作分配给任务以满足需求向量。我们研究需求向量随时间变化的问题的动态版本。在这里，目标是最小化切换成本，即响应需求向量变化而更改任务的代理数量。切换成本是一个重要的指标，因为更改任务可能会产生大量开销。我们研究了 Su、Su、Dornhaus 和 Lynch 引入的上述问题的数学形式化，这可以重新表述为寻找从对称差异到汉明距离的低失真嵌入的问题。在这个模型中，证明切换成本至少为 2 是微不足道的。我们通过为不同的参数范围给出 3 和 4 的下限，提出了切换成本的第一个非平凡下界。