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Documents authored by Boyd, Sylvia

Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.9
Approximation Algorithms for Flexible Graph Connectivity

Authors: Sylvia Boyd, Joseph Cheriyan, Arash Haddadan, and Sharat Ibrahimpur

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

Abstract
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.
Cite as

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}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.61
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, and Lu Wang

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

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.
Cite as

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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