## Abstract

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property π - e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property π. Furthermore, we study the on-line version of graph coloring whose offline version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Ω(n^{ε}) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Ω(n^{ε}) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

Original language | American English |
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Pages (from-to) | 354-393 |

Number of pages | 40 |

Journal | SIAM Journal on Computing |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

## Keywords

- Competitive analysis
- Graph problems
- Lower bounds
- Network optimization
- On-line computation
- Randomized algorithms