Robust subgraphs for trees and paths

Refael Hassin, Danny Segev

Research output: Contribution to journalArticlepeer-review


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 2n 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 languageEnglish
Pages (from-to)263-281
Number of pages19
JournalACM Transactions on Algorithms
Issue number2
StatePublished - 2006
Externally publishedYes


  • Approximation algorithms
  • Decomposition
  • Robust optimization
  • Spanning trees

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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