Speeding up Dynamic Programming in the Line-Constrained k-median

Abstract

In the planar k-median problem we are given a set of demand points and want to open up to k facilities as to minimize the sum of the transportation costs from each demand point to its nearest facility. In the line-constrained version the medians are required to lie on a given line. We present a new dynamic programming formulation for this problem, based on constructing a weighted DAG over a set of median candidates. We prove that, for any convex distance metric and any line, this DAG satisfies the concave Monge property. This allows us to construct efficient algorithms in $L_\infty$ and $L_1$ and any line, while the previously known solution (Wang and Zhang, ISAAC 2014) works only for vertical lines. We also provide an asymptotically optimal $\mathcal {O}(n)$ solution for the case of $k=1$.