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A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case

Authors Sylvia Boyd, Joseph Cheriyan, Robert Cummings, Logan Grout, Sharat Ibrahimpur , Zoltán Szigeti, Lu Wang

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Routing and network design problems
Keywords
  • 2-Edge Connectivity
  • Approximation Algorithms
  • Subtour LP for TSP

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Abstract

Given a connected undirected graph G ̅ on n vertices, and non-negative edge costs c, the 2ECM problem is that of finding a 2-edge connected spanning multisubgraph of G ̅ of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of G ̅, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution x, Carr and Ravi (1998) showed that the integrality gap is at most 4/3: they show that the vector 4/3 x dominates a convex combination of incidence vectors of 2-edge connected spanning multisubgraphs of G ̅.
We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lovász’s splitting-off theorem. Our proof naturally leads to a 4/3-approximation algorithm for half-integral instances. Given a half-integral solution x to the LP for 2ECM, we give an O(n²)-time algorithm to obtain a 2-edge connected spanning multisubgraph of G ̅ whose cost is at most 4/3 c^T x.

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Sylvia Boyd, Joseph Cheriyan, Robert Cummings, Logan Grout, Sharat Ibrahimpur, Zoltán Szigeti, and Lu Wang. A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.61

Author Details

Sylvia Boyd
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Joseph Cheriyan
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Robert Cummings
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Logan Grout
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Sharat Ibrahimpur
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Zoltán Szigeti
  • University Grenoble Alpes, CNRS, G-SCOP, France
Lu Wang
  • Department of Combinatorics and Optimization, University of Waterloo, Canada

Funding

  • Cheriyan, Joseph: Supported by NSERC RGPIN-2019-04197.
  • Ibrahimpur, Sharat: Supported in part by NSERC grant 327620-09.

Acknowledgements

{SB}, {JC}, and {SI} thank BIRS, Canada for organizing the workshop on The Traveling Salesman Problem (2018), where some of the results in our references were presented.

References

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