Remotely Useful Greedy Algorithms

Abstract

We study a class of parameterized $\max$-$\min$ problems, called Remote-$\mathcal {P}$: Given a minimization graph problem $\mathcal {P}$, find k vertices such that the optimum value of $\mathcal {P}$ is the highest amongst all k-node subsets. One simple example for Remote-$\mathcal {P}$ is computing the graph diameter where $\mathcal {P}$ is the shortest path problem and $k=2$. In this paper we focus on variants of the minimum spanning tree problem for $\mathcal {P}$. In previous literature $\mathcal {P}$ had to be defined on complete graphs. For many practically relevant problems it is natural to define $\mathcal {P}$ on sparse graphs, such as street networks. However, for large networks first computing the complete version of the network is impractical. Therefore, we describe greedy algorithms for Remote-$\mathcal {P}$ that perform well while computing only a small amount of shortest paths. On the theoretical side we proof a constant factor approximation. Furthermore, we implement and test the algorithms on a variety of graphs. We observe that the resulting running times are practical and that the quality is partially even better than the theoretical approximation guarantee, as shown via instance-based upper bounds.