## Abstract

We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.

Original language | English |
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Pages (from-to) | 143-162 |

Number of pages | 20 |

Journal | Networks and Heterogeneous Media |

Volume | 5 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Mar 2010 |

## Keywords

- Heat equation
- Integer programming
- Shortest path