## Abstract

In the traditional maximum-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow into outgoing edges. The problem is one of the most fundamental problems in TCS, with application in numerous domains. The fact a maximal-flow algorithm directs the flow in all the vertices of the network corresponds to a setting in which the authority has control in all vertices. Many applications in which the maximal-flow problem is applied involve an adversarial setting, where the authority does not have such a control. We introduce and study the unfortunate flow problem, which studies the flow that is guaranteed to reach the target when the edges that leave the source are saturated, yet the most unfortunate decisions are taken in the vertices. When the incoming flow to a vertex is greater than the outgoing capacity, flow is lost. The problem models evacuation scenarios where tra c is stuck due to jams in junctions and communication networks where packets are dropped in overloaded routers. We study the theoretical properties of unfortunate flows, show that the unfortunate-flow problem is co-NP-complete and point to polynomial fragments. We introduce and study interesting variants of the problem: integral unfortunate flow, where the flow along edges must be integral, controlled unfortunate flow, where the edges from the source need not be saturated and may be controlled, and no-loss controlled unfortunate flow, where the controlled flow must not be lost.

Original language | American English |
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Title of host publication | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |

Editors | Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella |

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

ISBN (Electronic) | 9783959770767 |

DOIs | |

State | Published - 1 Jul 2018 |

Event | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic Duration: 9 Jul 2018 → 13 Jul 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 107 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 9/07/18 → 13/07/18 |

### Bibliographical note

Publisher Copyright:© 2018 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

## Keywords

- Flow network
- Games
- Graph algorithms