## 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 online virtual circuit and optical routing problems. Lund and Yannakakis [LY93a] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any non-trivial, hereditary, property π. E.g., independent set, planar, acyclic, bipartite, etc. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H 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 off-line version has also been shown to be inapproximable [LY93b], on-line max edgedisjoint paths and on-line path coloring problems. Irrespective of the time complexity, we show an Ω,(n^{∈}) lower bound on the competitive ratio of randomized online algorithms for any of these problems. As a consequence, we obtain an Ω(n^{∈}) 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. Moreover, this lower bound holds even if the use of preemption is allowed. Similar lower bounds are obtained for on-line optical routing as well.

Original language | American English |
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Title of host publication | Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |

Publisher | Association for Computing Machinery |

Pages | 531-540 |

Number of pages | 10 |

ISBN (Electronic) | 0897917855 |

DOIs | |

State | Published - 1 Jul 1996 |

Externally published | Yes |

Event | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 - Philadelphia, United States Duration: 22 May 1996 → 24 May 1996 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F129452 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |
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Country/Territory | United States |

City | Philadelphia |

Period | 22/05/96 → 24/05/96 |

### Bibliographical note

Funding Information:*International Computer Science institute, Berkeley. Research supported in part by the Rothchild Postdoctoral fellowship. e-mail: yairb@icsi.berkeley. edu t DePWtment of Computer Science, Tel Aviv University, Aviv. Research supported in part by two grants from the Israel Academy of Sciences. e-mail: fiattlmath.tau.ac.il t International Computer Science Institute (Berkeley) & Di-partimento di Informatica Sistemistica, University di Roma “La Sapienza”. This work is partly supported by EU ESPRIT Long Term Research Project ALCOM-IT under contract n 20244, and by Italian Ministry of Scientific Research Project 40~o “Al-goritmi, Modelli di CHcolo e Strutture Informative”. e-mail: leOn@ldis.uniromal .it

Publisher Copyright:

© 1996 ACM.