Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

Authors Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, Andrew van der Poel



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Erik D. Demaine
  • MIT, Cambridge, MA, USA
Timothy D. Goodrich
  • NC State University, Raleigh, NC, USA
Kyle Kloster
  • NC State University, Raleigh, NC, USA
Brian Lavallee
  • NC State University, Raleigh, NC, USA
Quanquan C. Liu
  • MIT, Cambridge, MA, USA
Blair D. Sullivan
  • NC State University, Raleigh, NC, USA
Ali Vakilian
  • MIT, Cambridge, MA, USA
Andrew van der Poel
  • NC State University, Raleigh, NC, USA

Acknowledgements

We thank Abida Haque and Adam Hesterberg for helpful initial discussions, Nicole Wein for providing helpful references on bounded degeneracy problems, and Michael O'Brien for helpful comments on the manuscript.

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Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, and Andrew van der Poel. Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.37

Abstract

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set.
To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • structural rounding
  • graph editing
  • approximation algorithms

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