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Improved Approximation Algorithms for Capacitated Network Design and Flexible Graph Connectivity

Authors Ishan Bansal , Joe Cheriyan , Sanjeev Khanna , Miles Simmons

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • Capacitated network design
  • Covering small cuts
  • Edge-connectivity of graphs
  • f-Connectivity problem
  • Flexible Graph Connectivity
  • Knapsack-cover inequalities

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Abstract

We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity.
In the Cap-k-ECSS problem, we are given a graph G = (V,E) whose edges have non-negative costs and positive integer capacities, and the goal is to find a minimum-cost edge-set F such that every non-trivial cut of the graph G' = (V,F) has capacity at least k. Let n = |V| and let u_{min} (respectively, u_{max}) denote the minimum (respectively, maximum) capacity of an edge; assume that u_{max} ≤ k. We present an O(log({k}/u_{min}))-approximation algorithm for the Cap-k-ECSS problem, asymptotically improving upon the previous best approximation ratio of min(O(log{n}), k, 2u_{max}, 6 ⋅ {⌈ k/u_{min} ⌉}) whenever log(k/u_{min}) = o(log{n}) and u_{max} is sufficiently large.
In the (p,q)-Flexible Graph Connectivity problem, denoted (p,q)-FGC, the input is a graph G = (V, E) where E is partitioned into safe and unsafe edges, and the goal is to find a minimum-cost edge-set F such that the subgraph G' = (V, F) remains p-edge connected upon removal of any q unsafe edges from F. We present an 8-approximation algorithm for the (1,q)-FGC problem that improves upon the previous best approximation ratio of (q+1).
Both of our results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then, during the rounding process, utilizing a recent O(1)-approximation algorithm for the Cover Small Cuts problem. In the latter problem, the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than a specified value is covered by a link. We also show that the problem of covering small cuts inherently arises in another variant of (p,q)-FGC. Specifically, we give Cook reductions that preserve approximation ratios within O(1) factors between the (2,q)-FGC problem and the 2-Cover Small Cuts problem; in the latter problem, each small cut needs to be covered by two links.

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Ishan Bansal, Joe Cheriyan, Sanjeev Khanna, and Miles Simmons. Improved Approximation Algorithms for Capacitated Network Design and Flexible Graph Connectivity. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.20

Author Details

Ishan Bansal
  • Amazon, Bellevue, WA, USA
Joe Cheriyan
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Sanjeev Khanna
  • Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA
Miles Simmons
  • Department of Combinatorics and Optimization, University of Waterloo, Canada

Funding

  • Bansal, Ishan: This work is external and does not relate to the position at Amazon.
  • Cheriyan, Joe: Supported in part by NSERC grant RGPIN-2024-04473.
  • Khanna, Sanjeev: Supported in part by NSF award CCF-2402284 and AFOSR award FA9550-25-1-0107.

Acknowledgements

We thank the anonymous reviewers and the ICALP PC for their comments. We are grateful to David Aleman Espinosa and Sharat Ibrahimpur for reading a preliminary version, and for their detailed comments.

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