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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

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.

Ishan Bansal, Joseph Cheriyan, Logan Grout, and Sharat Ibrahimpur. Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 15:1-15:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@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/entities/document/10.4230/LIPIcs.ICALP.2023.15}, 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} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We present a 5/3-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost.
A 7/4-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a 7/4-approximation algorithm," Math. Program., 182(1):315-354, 2020.
Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

Joseph Cheriyan, Robert Cummings, Jack Dippel, and Jasper Zhu. An Improved Approximation Algorithm for the Matching Augmentation Problem. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 38:1-38:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cheriyan_et_al:LIPIcs.ISAAC.2021.38, author = {Cheriyan, Joseph and Cummings, Robert and Dippel, Jack and Zhu, Jasper}, title = {{An Improved Approximation Algorithm for the Matching Augmentation Problem}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {38:1--38:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.38}, URN = {urn:nbn:de:0030-drops-154714}, doi = {10.4230/LIPIcs.ISAAC.2021.38}, annote = {Keywords: 2-Edge connected graph, 2-edge covers, approximation algorithms, connectivity augmentation, forest augmentation problem, matching augmentation problem, network design} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and Mühlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), IPCO 2020: pp. 13-26). In an instance of the Flexible Graph Connectivity (FGC) problem, we have an undirected connected graph G = (V,E), a partition of E into a set of safe edges S and a set of unsafe edges U, and nonnegative costs {c_e}_{e ∈ E} on the edges. A subset F ⊆ E of edges is feasible for FGC if for any unsafe edge e ∈ F ∩ U, the subgraph (V,F⧵{e}) is connected. The algorithmic goal is to find a (feasible) solution F that minimizes c(F) = ∑_{e ∈ F} c_e. We present a simple 2-approximation algorithm for FGC via a reduction to the minimum-cost r-out 2-arborescence problem. This improves upon the 2.527-approximation algorithm of Adjiashvili et al.
For integers p ≥ 1 and q ≥ 0, the (p,q)-FGC problem is a generalization of FGC where we seek a minimum-cost subgraph H = (V,F) that remains p-edge connected against the failure of any set of at most q unsafe edges; that is, for any set F' ⊆ U with |F'| ≤ q, H-F' = (V, F ⧵ F') should be p-edge connected. Note that FGC corresponds to the (1,1)-FGC problem. We give approximation algorithms for two important special cases of (p,q)-FGC: (a) Our 2-approximation algorithm for FGC extends to a (k+1)-approximation algorithm for the (1,k)-FGC problem. (b) We present a 4-approximation algorithm for the (k,1)-FGC problem.
For the unweighted FGC problem, where each edge has unit cost, we give a 16/11-approximation algorithm. This improves on the result of Adjiashvili et al. for this problem.
The (p,q)-FGC model with p = 1 or q ≤ 1 can be cast as the Capacitated k-Connected Subgraph problem which is a special case of the well-known Capacitated Network Design problem. We denote the former problem by Cap-k-ECSS. An instance of this problem consists of an undirected graph G = (V,E), nonnegative integer edge-capacities {u_e}_{e ∈ E}, nonnegative edge-costs {c_e}_{e ∈ E}, and a positive integer k. The goal is to find a minimum-cost edge-set F ⊆ E such that every (non-trivial) cut of the capacitated subgraph H(V,F,u) has capacity at least k. We give a min(k, 2max_{e ∈ E} u_e)-approximation algorithm for this problem.

Sylvia Boyd, Joseph Cheriyan, Arash Haddadan, and Sharat Ibrahimpur. Approximation Algorithms for Flexible Graph Connectivity. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{boyd_et_al:LIPIcs.FSTTCS.2021.9, author = {Boyd, Sylvia and Cheriyan, Joseph and Haddadan, Arash and Ibrahimpur, Sharat}, title = {{Approximation Algorithms for Flexible Graph Connectivity}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.9}, URN = {urn:nbn:de:0030-drops-155206}, doi = {10.4230/LIPIcs.FSTTCS.2021.9}, annote = {Keywords: Approximation Algorithms, Combinatorial Optimization, Network Design, Edge-Connectivity of Graphs, Reliability of Networks} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

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.

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)

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@InProceedings{boyd_et_al:LIPIcs.APPROX/RANDOM.2020.61, author = {Boyd, Sylvia and Cheriyan, Joseph and Cummings, Robert and Grout, Logan and Ibrahimpur, Sharat and Szigeti, Zolt\'{a}n and Wang, Lu}, title = {{A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {61:1--61:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.61}, URN = {urn:nbn:de:0030-drops-126643}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.61}, annote = {Keywords: 2-Edge Connectivity, Approximation Algorithms, Subtour LP for TSP} }

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