Various real-world problems consist of partitioning a set of locations into disjoint subsets, each subset spread in a way that it covers the whole set with a certain radius. Given a finite set S, a metric d, and a radius r, define a subset (of S) S' to be an r-cover if and only if forall s in S there exists s' in S' such that d(s,s') is less or equal to r. We examine the problem of determining whether there exist m disjoint r-covers, naming it the Solidarity Cover Problem (SCP). We consider as well the related optimization problems of maximizing the number of r-covers, referred to as the partition size, and minimizing the radius. We analyze the relation between the SCP and a graph problem known as the Domatic Number Problem (DNP), both hard problems in the general case. We show that the SCP is hard already in the Euclidean 2D setting, implying hardness of the DNP already in the unit-disc-graph setting. As far as we know, the latter is a result yet to be shown. We use the tight approximation bound of (1-o(1))/ln(n) for the DNP's general case, shown by U.Feige, M.Halld'orsson, G.Kortsarz, and A.Srinivasan (SIAM Journal on computing, 2002), to deduce the same bound for partition-size approximation of the SCP in the Euclidean space setting. We show an upper bound of 3 and lower bounds of 2 and sqrt(2) for approximating the minimal radius in different settings of the SCP. Lastly, in the Euclidean 2D setting we provide a general bicriteria-approximation scheme which allows a range of possibilities for trading the optimality of the radius in return for better approximation of the partition size and vice versa. We demonstrate a usage of the scheme which achieves an approximation of (1/16,2) for the partition size and radius respectively.

翻译：各种现实问题涉及将一组位置划分为不相交的子集，每个子集的分布需确保以特定半径覆盖整个集合。给定有限集合S、度量d和半径r，定义子集S'⊆S为r-覆盖当且仅当对任意s∈S存在s'∈S'满足d(s,s') ≤ r。本文研究了判定是否存在m个互不相交的r-覆盖的问题，并将其命名为固覆盖问题（SCP）。同时考虑相关优化问题：最大化r-覆盖数量（即划分规模）以及最小化半径。我们分析了SCP与图论中的支配集数问题（DNP）之间的关联，两者在一般情况下均为难解问题。研究证明SCP在二维欧几里得空间中已具难解性，进而推导出DNP在单位圆盘图场景下的难解性——据我们所知，该结论此前尚未被证明。我们利用U.Feige、M.Halldórsson、G.Kortsarz和A.Srinivasan（SIAM计算期刊，2002）给出的DNP一般情况下的紧逼近界(1-o(1))/ln(n)，推导出欧几里得空间下SCP划分规模逼近的相同界值。针对SCP不同场景下的最小半径逼近问题，我们给出上界3以及下界2和√2。最后，在二维欧几里得场景中提出通用双准则逼近方案，允许在半径最优性与划分规模逼近质量之间进行权衡，并展示了该方案的一个应用实例，其对于划分规模和半径分别达到(1/16,2)的逼近效果。