## Abstract

The Distance Realization problem is defined as follows. Given an n × n matrix D of nonnegative integers, interpreted as inter-vertex distances, find an n-vertex weighted or unweighted graph G realizing D, i.e., whose inter-vertex distances satisfy distG(i, j) = Di,j for every 1 ≤ i < j ≤ n, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of Distance Realization that was studied in the past is where each entry in the matrix D may contain a range of consecutive permissible values. We refer to this extension as Range Distance Realization (or Range-DR). Restricting each range to at most k values yields the problem k-Range Distance Realization (or k-Range-DR). The current paper introduces a new extension of Distance Realization, in which each entry Di,j of the matrix may contain an arbitrary set of acceptable values for the distance between i and j, for every 1 ≤ i < j ≤ n. We refer to this extension as Set Distance Realization (Set-DR), and to the restricted problem where each entry may contain at most k values as k-Set Distance Realization (or k-Set-DR). We first show that 2-Range-DR is NP-hard for unweighted graphs (implying the same for 2-Set-DR). Next we prove that 2-Set-DR is NP-hard for unweighted and weighted trees. We then explore Set-DR where the realization is restricted to the families of stars, paths, or cycles. For the weighted case, our positive results are that for each of these families there exists a polynomial time algorithm for 2-Set-DR. On the hardness side, we prove that 6-Set-DR is NP-hard for stars and 5-Set-DR is NP-hard for paths and cycles. For the unweighted case, our results are the same, except for the case of unweighted stars, for which k-Set-DR is polynomially solvable for any k.

Original language | English |
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Title of host publication | 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022 |

Editors | Stefan Szeider, Robert Ganian, Alexandra Silva |

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

ISBN (Electronic) | 9783959772563 |

DOIs | |

State | Published - 1 Aug 2022 |

Externally published | Yes |

Event | 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022 - Vienna, Austria Duration: 22 Aug 2022 → 26 Aug 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 241 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022 |
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Country/Territory | Austria |

City | Vienna |

Period | 22/08/22 → 26/08/22 |

## Keywords

- Graph Realization
- distance realization
- network design