The Solidarity Cover Problem
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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.
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