On The Hardness Of Range Assignment Problems

摘要

We investigate the computational hardness of the connectivity, the strong connectivity, and the broadcast type of range assignment problems in R~2 and R~3. We present new reductions for the connectivity problem, which are easily adapted to suit the other two problems. All reductions are considerably simpler than the technically quite involved ones used in earlier works on these problems. Using our constructions, we can for the first time prove NP-hardness of these problems for all real distance-power gradients α > 0 (resp. α > 1 for broadcast) in 2D, and prove APX-hardness of all three problems in 3D for all α > 1. Our reductions yield improved lower bounds on the approximation ratios for all problems where APX-hardness was known before already. In particular, we derive the overall first APX-hardness proof for broadcast. This was an open problem posed in earlier work in this area, as was the question whether (strong) connectivity remains NP-hard for α = 1. In addition, we give the first hardness results for so-called well-spread instances. On the positive side, we prove that two natural greedy algorithms are 2-approximations for (strong) connectivity, and show that the factor 2 is tight in R~2 for α > 1. We also analyze the performance guarantee of the well-known MST-heuristic as a function of the input size.