## Abstract

Motivated by the desire to utilize a limited number of configurable optical switches by recent advances in Software Defined Networks (SDNs), we define an online problem which we call the Caching in Matchings problem. This problem has a natural combinatorial structure and therefore may find additional applications in theory and practice. In the Caching in Matchings problem our cache consists of k matchings of connections between servers that form a bipartite graph. To cache a connection we insert it into one of the k matchings possibly evicting at most two other connections from this matching. This problem resembles the problem known as Connection Caching [20], where we also cache connections but our only restriction is that they form a graph with bounded degree k. Our results show a somewhat surprising qualitative separation between the problems: The competitive ratio of any online algorithm for caching in matchings must depend on the size of the graph. Specifically, we give a deterministic O(nk) competitive and randomized O(n log k) competitive algorithms for caching in matchings, where n is the number of servers and k is the number of matchings. We also show that the competitive ratio of any deterministic algorithm is Ω(max(^{n}_{k} , k)) and of any randomized algorithm is Ω(log_{k2 log}^{n}_{k} ·log k). In particular, the lower bound for randomized algorithms is Ω(log n) regardless of k, and can be as high as Ω(log^{2} n) if k = n^{1}/^{3}, for example. We also show that if we allow the algorithm to use at least 2k − 1 matchings compared to k used by the optimum then we match the competitive ratios of connection catching which are independent of n. Interestingly, we also show that even a single extra matching for the algorithm allows to get substantially better bounds.

Original language | English |
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Title of host publication | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |

Editors | Karl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773225 |

DOIs | |

State | Published - Jul 2024 |

Event | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia Duration: 8 Jul 2024 → 12 Jul 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 297 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |
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Country/Territory | Estonia |

City | Tallinn |

Period | 8/07/24 → 12/07/24 |

## Keywords

- Caching
- Caching in Matchings
- Edge Coloring
- Matchings
- Online Algorithms