Abstract
We characterize the complexity of several basic optimization problems for unit disk graphs specified hierarchically as in [LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered. These problems include minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum independent dominating set.
Our PSPACE-hardness results imply the PSPACE-hardness of the geometric location problems in [MS84, WK88], when sets of points are specified hierarchically as in [BOW83] or [LW92]. Also, each of our PSPACE-hardness results holds, when the hierarchical specifications are 1-level restricted (Definition 2.2) and the graphs are specified hierarchically either as in [BOW83] or as in [LW92].
For k-level restricted hierarchical specifications, where k is fixed, we have also developed a polynomial time algorithm to solve the maximum clique problem, and a polynomial time relative approximation algorithm for minimum coloring.
This research was supported by NSF Grants CCR-89-03319 and CCR-90-06396.
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Marathe, M.V., Radhakrishnan, V., Hunt, H.B., Ravi, S.S. (1994). Hierarchically specified unit disk graphs. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1993. Lecture Notes in Computer Science, vol 790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57899-4_38
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