Fixed-Parameter Approximations for k-Center Problems in Low Highway Dimension Graphs

Abstract

We consider the k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex of G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. (SODA, pp 782–793, 2010). We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any $${\varepsilon }>0$$ computing a $$(2-{\varepsilon })$$ -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension $$O(\log ^2 n)$$ . The latter does not rule out fixed-parameter $$(2-{\varepsilon })$$ -approximations for the highway dimension parameter h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless ETH fails. On the positive side, we show how to get below the approximation factor of 2 by combining the parameters k and h: we develop a fixed-parameter 3 / 2-approximation with running time $$2^{O(kh\log h)}\cdot n^{O(1)}$$ . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter $$(2-{\varepsilon })$$ -approximations for only the parameter h. We also provide similar fixed-parameter approximations for the weightedk-Center and $$(k,{\mathcal {F}})$$ -Partition problems, which generalize k-Center.