Non-greedy Online Steiner Trees on Outerplanar Graphs

Abstract

This paper addresses the classical online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we did not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is \(\varOmega (\log n)\)-competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. The algorithm can be applied to a 21.752-competitive file allocation algorithm against adaptive online adversaries on outerplanar graphs. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.