License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.15
URN: urn:nbn:de:0030-drops-180678
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18067/
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Bansal, Ishan ; Cheriyan, Joseph ; Grout, Logan ; Ibrahimpur, Sharat

Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

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Abstract

We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al. prove an approximation ratio of two for connectivity augmentation problems where the connectivity requirements can be specified by uncrossable functions. They state: "Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar... A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems."
Our main result proves a 16-approximation ratio via the primal-dual method for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions have a laminar support.
We present applications of our main result to three network-design problems.
1) A 16-approximation algorithm for augmenting the family of small cuts of a graph G. The previous best approximation ratio was O(log |V(G)|).
2) A 16⋅⌈k/u_min⌉-approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph G = (V,E) with edge costs c ∈ ℚ_{≥0}^E and edge capacities u ∈ ℤ_{≥0}^E, find a minimum cost subset of the edges F ⊆ E such that the capacity across any cut in (V,F) is at least k; u_min (respectively, u_max) denote the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. u_max ≤ k. The previous best approximation ratio was min(O(log|V|), k, 2u_max).
3) A 20-approximation algorithm for the model of (p,2)-Flexible Graph Connectivity. The previous best approximation ratio was O(log|V(G)|), where G denotes the input graph.

BibTeX - Entry

@InProceedings{bansal_et_al:LIPIcs.ICALP.2023.15,
  author =	{Bansal, Ishan and Cheriyan, Joseph and Grout, Logan and Ibrahimpur, Sharat},
  title =	{{Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18067},
  URN =		{urn:nbn:de:0030-drops-180678},
  doi =		{10.4230/LIPIcs.ICALP.2023.15},
  annote =	{Keywords: Approximation algorithms, Edge-connectivity of graphs, f-Connectivity problem, Flexible Graph Connectivity, Minimum cuts, Network design, Primal-dual method, Small cuts}
}

Keywords: Approximation algorithms, Edge-connectivity of graphs, f-Connectivity problem, Flexible Graph Connectivity, Minimum cuts, Network design, Primal-dual method, Small cuts
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023


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