## Abstract

Consider a graph problem which is associated with a parameter, for example, that of finding a longest tour spanning k vertices. The following question is natural: Is there a small subgraph that contains an optimal or near optimal solution for every possible value of the given parameter? Such a subgraph is said to be robust. In this article we consider the problems of finding heavy paths and heavy trees of k edges. In these two cases, we prove surprising bounds on the size of a robust subgraph for a variety of approximation ratios. For both problems, we show that in every complete weighted graph on n vertices there exists a subgraph with approximately α/1-α ^{2}n edges that contains an α-approximate solution for every k = 1,..., n - 1. In the analysis of the tree problem, we also describe a new result regarding balanced decomposition of trees. In addition, we consider variants in which the subgraph itself is restricted to be a path or a tree. For these problems, we describe polynomial time algorithms and corresponding proofs of negative results.

Original language | English |
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Pages (from-to) | 263-281 |

Number of pages | 19 |

Journal | ACM Transactions on Algorithms |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

## Keywords

- Approximation algorithms
- Decomposition
- Robust optimization
- Spanning trees

## ASJC Scopus subject areas

- Mathematics (miscellaneous)