On the Complexity of Partitioning Graphs for Arc-Flags

Abstract

Precomputation of auxiliary data in an additional off-line step is a common approach towards improving the performance of shortest-path queries in large-scale networks. One such technique is the arc-flags algorithm, where the preprocessing involves computing a partition of the input graph. The quality of this partition significantly affects the speed-up observed in the query phase. It is evaluated by considering the search-space size of subsequent shortest-path queries, in particular its maximum or its average over all queries. In this paper, we substantially strengthen existing hardness results of Bauer et al. and show that optimally filling this degree of freedom is N P-hard for trees with unit-length edges, even if we bound the height or the degree. On the other hand, we show that optimal partitions for paths can be computed efficiently and give approximation algorithms for cycles and trees. ACM Subject Classification G.2.2 Graph Theory

Route Planning in Transportation Networks

“…Deciding which k shortcuts (for fixed k) to add to a graph in order to minimize the SHARC search space is also NP-hard [38]. Bauer et al [35] also analyze the preprocessing of Arc Flags in more detail and on restricted graph classes, such as paths, trees, and cycles, and show that finding an optimal partition is NP-hard even for binary trees.…”