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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Fairness considerations have motivated new clustering problems and algorithms in recent years. In this paper we consider the Priority Matroid Median problem which generalizes the Priority k-Median problem that has recently been studied. The input consists of a set of facilities ℱ and a set of clients 𝒞 that lie in a metric space (ℱ ∪ 𝒞,d), and a matroid ℳ = (ℱ,ℐ) over the facilities. In addition, each client j has a specified radius r_j ≥ 0 and each facility i ∈ ℱ has an opening cost f_i > 0. The goal is to choose a subset S ⊆ ℱ of facilities to minimize ∑_{i ∈ ℱ} f_i + ∑_{j ∈ 𝒞} d(j,S) subject to two constraints: (i) S is an independent set in ℳ (that is S ∈ ℐ) and (ii) for each client j, its distance to an open facility is at most r_j (that is, d(j,S) ≤ r_j). For this problem we describe the first bicriteria (c₁,c₂) approximations for fixed constants c₁,c₂: the radius constraints of the clients are violated by at most a factor of c₁ and the objective cost is at most c₂ times the optimum cost. We also improve the previously known bicriteria approximation for the uniform radius setting (r_j : = L ∀ j ∈ 𝒞).

Tanvi Bajpai and Chandra Chekuri. Bicriteria Approximation Algorithms for Priority Matroid Median. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 7:1-7:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bajpai_et_al:LIPIcs.APPROX/RANDOM.2023.7, author = {Bajpai, Tanvi and Chekuri, Chandra}, title = {{Bicriteria Approximation Algorithms for Priority Matroid Median}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {7:1--7:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.7}, URN = {urn:nbn:de:0030-drops-188328}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.7}, annote = {Keywords: k-median, fair clustering, matroid} }

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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Maximum weight independent set (MWIS) admits a 1/k-approximation in inductively k-independent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)-approximation in k-perfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize k-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others [Ye and Borodin, 2012; Kammer and Tholey, 2014]. We consider a generalization of MWIS to a submodular objective. Given a graph G = (V,E) and a non-negative submodular function f: 2^V → ℝ_+, the goal is to approximately solve max_{S ∈ ℐ_G} f(S) where ℐ_G is the set of independent sets of G. We obtain an Ω(1/k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1/e(k+1). This approach also yields parallel (or low-adaptivity) approximations.
Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively k-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

Chandra Chekuri and Kent Quanrud. Independent Sets in Elimination Graphs with a Submodular Objective. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 24:1-24:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2023.24, author = {Chekuri, Chandra and Quanrud, Kent}, title = {{Independent Sets in Elimination Graphs with a Submodular Objective}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {24:1--24:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.24}, URN = {urn:nbn:de:0030-drops-188490}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.24}, annote = {Keywords: elimination graphs, independent set, submodular maximization, primal-dual} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Boob et al. [Boob et al., 2020] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [Chandra Chekuri et al., 2022] extended the algorithm to supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige’s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [Satoru Fujishige, 1980], and a noisy version of the Frank-Wolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. This yields a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige’s result and convex optimization.

Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 56:1-56:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harb_et_al:LIPIcs.ESA.2023.56, author = {Harb, Elfarouk and Quanrud, Kent and Chekuri, Chandra}, title = {{Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.56}, URN = {urn:nbn:de:0030-drops-187091}, doi = {10.4230/LIPIcs.ESA.2023.56}, annote = {Keywords: Polymatroid, lexicographically optimum base, densest subgraph, tree packing} }

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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)

Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model [Adjiashvili, 2013; Adjiashvili et al., 2020; Adjiashvili et al., 2022; Boyd et al., 2023], in which the edge set is partitioned into safe and unsafe edges. Given parameters p,q ≥ 1, the goal is to find a cheap subgraph that remains p-connected even after the failure of q unsafe edges. We also discuss the bulk-robust model [Adjiashvili et al., 2015; Adjiashvili, 2015] and the relative survivable network design model [Dinitz et al., 2022]. While SNDP admits a 2-approximation [K. Jain, 2001], the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions.
Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when p and q are fixed, we consider two special cases. For the s-t case we obtain an approximation ratio that depends only on p,q whenever p+q > pq/2 which includes (p,2) and (2,q) for all p,q ≥ 1. For the global connectivity case we obtain an O(q) approximation for (2,q), and an O(p) approximation for (p,2) and (p,3) for any p ≥ 1, and for (p,4) when p is even. These are based on an augmentation framework and decomposing the families of cuts that need to be covered into a small number of uncrossable families.
Our second result is a poly-logarithmic approximation for a generalization of the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. We utilize a recent framework due to Chen et al. [Chen et al., 2022] that was designed for handling group connectivity.

Chandra Chekuri and Rhea Jain. Approximation Algorithms for Network Design in Non-Uniform Fault Models. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 36:1-36:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2023.36, author = {Chekuri, Chandra and Jain, Rhea}, title = {{Approximation Algorithms for Network Design in Non-Uniform Fault Models}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {36:1--36:20}, 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.2023.36}, URN = {urn:nbn:de:0030-drops-180885}, doi = {10.4230/LIPIcs.ICALP.2023.36}, annote = {Keywords: non-uniform faults, network design, approximation algorithm} }

Document

**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Submodular functions are fundamental to combinatorial optimization. Many interesting problems can be formulated as special cases of problems involving submodular functions. In this work, we consider the problem of approximating symmetric submodular functions everywhere using hypergraph cut functions. Devanur, Dughmi, Schwartz, Sharma, and Singh [Devanur et al., 2013] showed that symmetric submodular functions over n-element ground sets cannot be approximated within (n/8)-factor using a graph cut function and raised the question of approximating them using hypergraph cut functions. Our main result is that there exist symmetric submodular functions over n-element ground sets that cannot be approximated within a o(n^{1/3}/log² n)-factor using a hypergraph cut function. On the positive side, we show that symmetrized concave linear functions and symmetrized rank functions of uniform matroids and partition matroids can be constant-approximated using hypergraph cut functions.

Calvin Beideman, Karthekeyan Chandrasekaran, Chandra Chekuri, and Chao Xu. Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 6:1-6:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{beideman_et_al:LIPIcs.FSTTCS.2022.6, author = {Beideman, Calvin and Chandrasekaran, Karthekeyan and Chekuri, Chandra and Xu, Chao}, title = {{Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.6}, URN = {urn:nbn:de:0030-drops-173986}, doi = {10.4230/LIPIcs.FSTTCS.2022.6}, annote = {Keywords: Submodular Functions, Hypergraphs, Approximation, Representation} }

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Complete Volume

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

LIPIcs, Volume 213, FSTTCS 2021, Complete Volume

Mikołaj Bojańczyk and Chandra Chekuri. LIPIcs, Volume 213, FSTTCS 2021, Complete Volume. 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. 1-866, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@Proceedings{bojanczyk_et_al:LIPIcs.FSTTCS.2021, title = {{LIPIcs, Volume 213, FSTTCS 2021, Complete Volume}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {1--866}, 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}, URN = {urn:nbn:de:0030-drops-155102}, doi = {10.4230/LIPIcs.FSTTCS.2021}, annote = {Keywords: LIPIcs, Volume 213, FSTTCS 2021, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

Mikołaj Bojańczyk and Chandra Chekuri. Front Matter, Table of Contents, Preface, Conference Organization. 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. 0:i-0:xvi, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bojanczyk_et_al:LIPIcs.FSTTCS.2021.0, author = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {0:i--0:xvi}, 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.2021.0}, URN = {urn:nbn:de:0030-drops-155113}, doi = {10.4230/LIPIcs.FSTTCS.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a (1+ε) approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate B such that B ≤ B^* ≤ ⌈(1+ε)B⌉+1 where B^* is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.

Chandra Chekuri, Kent Quanrud, and Manuel R. Torres. Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 24:1-24:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2021.24, author = {Chekuri, Chandra and Quanrud, Kent and Torres, Manuel R.}, title = {{Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {24:1--24:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.24}, URN = {urn:nbn:de:0030-drops-147177}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.24}, annote = {Keywords: bounded degree spanning tree, crossing spanning tree, swap rounding, fast approximation algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

In the Priority k-Center problem, the input consists of a metric space (X,d), an integer k and for each point v ∈ X a priority radius r(v). The goal is to choose k-centers S ⊆ X to minimize max_{v ∈ X} 1/(r(v)) d(v,S). If all r(v)’s were uniform, one obtains the classical k-center problem. Plesník [Ján Plesník, 1987] introduced this problem and gave a 2-approximation algorithm matching the best possible algorithm for vanilla k-center. We show how the Priority k-Center problem is related to two different notions of fair clustering [Harris et al., 2019; Christopher Jung et al., 2020]. Motivated by these developments we revisit the problem and, in our main technical contribution, develop a framework that yields constant factor approximation algorithms for Priority k-Center with outliers. Our framework extends to generalizations of Priority k-Center to matroid and knapsack constraints, and as a corollary, also yields algorithms with fairness guarantees in the lottery model of Harris et al.

Tanvi Bajpai, Deeparnab Chakrabarty, Chandra Chekuri, and Maryam Negahbani. Revisiting Priority k-Center: Fairness and Outliers. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 21:1-21:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bajpai_et_al:LIPIcs.ICALP.2021.21, author = {Bajpai, Tanvi and Chakrabarty, Deeparnab and Chekuri, Chandra and Negahbani, Maryam}, title = {{Revisiting Priority k-Center: Fairness and Outliers}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {21:1--21:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.21}, URN = {urn:nbn:de:0030-drops-140909}, doi = {10.4230/LIPIcs.ICALP.2021.21}, annote = {Keywords: Fairness, Clustering, Approximation, Outliers} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time Õ(n²). For rooted edge connectivity this is the first algorithm to improve on the Ω(n³) time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems.
We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a (1+ε)-approximation for rooted vertex connectivity in Õ(nW/ε) time where W is the total vertex weight (assuming integral vertex weights); in particular this yields an Õ(n²/ε) time randomized algorithm for unweighted graphs. This translates to a Õ(KnW) time exact algorithm where K is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity.
Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [Harold N. Gabow, 1995] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [Nanongkai et al., 2019] and Forster et al. [Sebastian Forster et al., 2020] for vertex connectivity.

Chandra Chekuri and Kent Quanrud. Faster Algorithms for Rooted Connectivity in Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 49:1-49:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2021.49, author = {Chekuri, Chandra and Quanrud, Kent}, title = {{Faster Algorithms for Rooted Connectivity in Directed Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {49:1--49:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.49}, URN = {urn:nbn:de:0030-drops-141187}, doi = {10.4230/LIPIcs.ICALP.2021.49}, annote = {Keywords: rooted connectivity, directed graph, fast algorithm, sparsification} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations.
In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.

Chandra Chekuri and Kent Quanrud. Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 50:1-50:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2021.50, author = {Chekuri, Chandra and Quanrud, Kent}, title = {{Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {50:1--50:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.50}, URN = {urn:nbn:de:0030-drops-141199}, doi = {10.4230/LIPIcs.ICALP.2021.50}, annote = {Keywords: cuts, vertex connectivity, hypergraphs, fast algorithms, submodularity, bisumodularity, lattices, isolating cuts, element connectivity} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008].
In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008].

Chandra Chekuri, Kent Quanrud, and Chao Xu. LP Relaxation and Tree Packing for Minimum k-cuts. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 7:1-7:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chekuri_et_al:OASIcs.SOSA.2019.7, author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao}, title = {{LP Relaxation and Tree Packing for Minimum k-cuts}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {7:1--7:18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.7}, URN = {urn:nbn:de:0030-drops-100335}, doi = {10.4230/OASIcs.SOSA.2019.7}, annote = {Keywords: k-cut, LP relaxation, tree packing} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We consider clustering in the perturbation resilience model that has been studied since the work of Bilu and Linial [Yonatan Bilu and Nathan Linial, 2010] and Awasthi, Blum and Sheffet [Awasthi et al., 2012]. A clustering instance I is said to be alpha-perturbation resilient if the optimal solution does not change when the pairwise distances are modified by a factor of alpha and the perturbed distances satisfy the metric property - this is the metric perturbation resilience property introduced in [Angelidakis et al., 2017] and a weaker requirement than prior models. We make two high-level contributions.
- We show that the natural LP relaxation of k-center and asymmetric k-center is integral for 2-perturbation resilient instances. We belive that demonstrating the goodness of standard LP relaxations complements existing results [Maria{-}Florina Balcan et al., 2016; Angelidakis et al., 2017] that are based on new algorithms designed for the perturbation model.
- We define a simple new model of perturbation resilience for clustering with outliers. Using this model we show that the unified MST and dynamic programming based algorithm proposed in [Angelidakis et al., 2017] exactly solves the clustering with outliers problem for several common center based objectives (like k-center, k-means, k-median) when the instances is 2-perturbation resilient. We further show that a natural LP relxation is integral for 2-perturbation resilient instances of k-center with outliers.

Chandra Chekuri and Shalmoli Gupta. Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chekuri_et_al:LIPIcs.APPROX-RANDOM.2018.9, author = {Chekuri, Chandra and Gupta, Shalmoli}, title = {{Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.9}, URN = {urn:nbn:de:0030-drops-94136}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.9}, annote = {Keywords: Clustering, Perturbation Resilience, LP Integrality, Outliers, Beyond Worst Case Analysis} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

In this note we consider the survivable network design problem (SNDP) in undirected graphs. We make two contributions. The first is a new counting argument in the iterated rounding based 2-approximation for edge-connectivity SNDP (EC-SNDP) originally due to Jain. The second contribution is to make some connections between hypergraphic version of SNDP (Hypergraph-SNDP) introduced by Zhao, Nagamochi and Ibaraki, and edge and node-weighted versions of EC-SNDP and element-connectivity SNDP (Elem-SNDP). One useful consequence is a 2-approximation for Elem-SNDP that avoids the use of set-pair based relaxation and analysis.

Chandra Chekuri and Thapanapong Rukkanchanunt. A Note on Iterated Rounding for the Survivable Network Design Problem. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 2:1-2:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chekuri_et_al:OASIcs.SOSA.2018.2, author = {Chekuri, Chandra and Rukkanchanunt, Thapanapong}, title = {{A Note on Iterated Rounding for the Survivable Network Design Problem}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {2:1--2:10}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.2}, URN = {urn:nbn:de:0030-drops-82976}, doi = {10.4230/OASIcs.SOSA.2018.2}, annote = {Keywords: survivable network design, iterated rounding, approximation, element connectivity} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

Congestion minimization is a well-known routing problem for which
there is an O(log n/loglog n)-approximation via randomized rounding due to Raghavan and Thompson. Srinivasan formally introduced the low-congestion multi-path routing problem as a generalization of the (single-path) congestion minimization problem. The goal is to route multiple disjoint paths for each pair, for the sake of fault tolerance. Srinivasan developed a dependent randomized scheme for a special case of the multi-path problem when the input consists of a given set of disjoint paths for each pair and the goal is to select a given subset of them. Subsequently Doerr gave a different dependentrounding scheme and derandomization. Doerr et al. considered the problem where the paths have to be chosen, and applied the dependent rounding technique and evaluated it experimentally. However, their algorithm does not maintain the required disjointness property without which the problem easily reduces to the standard congestion minimization problem.
In this note we show a simple algorithm that solves the problem
correctly without the need for dependent rounding --- standard
independent rounding suffices. This is made possible via the notion
of multiroute flows originally suggested by Kishimoto et al. One advantage of the simpler rounding is an improved bound on the congestion when the path lengths are short.

Chandra Chekuri and Mark Idleman. Congestion Minimization for Multipath Routing via Multiroute Flows. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 3:1-3:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chekuri_et_al:OASIcs.SOSA.2018.3, author = {Chekuri, Chandra and Idleman, Mark}, title = {{Congestion Minimization for Multipath Routing via Multiroute Flows}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {3:1--3:12}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.3}, URN = {urn:nbn:de:0030-drops-82984}, doi = {10.4230/OASIcs.SOSA.2018.3}, annote = {Keywords: multipath routing, congestion minimization, multiroute flows} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h)).

Chandra Chekuri, Alina Ene, and Marcin Pilipczuk. Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 7:1-7:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2016.7, author = {Chekuri, Chandra and Ene, Alina and Pilipczuk, Marcin}, title = {{Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.7}, URN = {urn:nbn:de:0030-drops-62737}, doi = {10.4230/LIPIcs.ICALP.2016.7}, annote = {Keywords: Disjoint paths, symmetric demands, planar directed graph} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Given an edge-weighted undirected graph $G$ with a specified set of
terminals, let the \emph{density} of any subgraph be the ratio of
its weight/cost to the number of terminals it contains. If $G$ is
2-connected, does it contain smaller 2-connected subgraphs of
density comparable to that of $G$? We answer this question in the
affirmative by giving an algorithm to \emph{prune} $G$ and find such
subgraphs of any desired size, at the cost of only a logarithmic
increase in density (plus a small additive factor).
We apply the pruning techniques to give algorithms for two NP-Hard
problems on finding large 2-vertex-connected subgraphs of low cost;
no previous approximation algorithm was known for either problem. In
the \kv problem, we are given an undirected graph $G$ with edge
costs and an integer $k$; the goal is to find a minimum-cost
2-vertex-connected subgraph of $G$ containing at least $k$
vertices. In the \bv\ problem, we are given the graph $G$ with edge
costs, and a budget $B$; the goal is to find a 2-vertex-connected
subgraph $H$ of $G$ with total edge cost at most $B$ that maximizes
the number of vertices in $H$. We describe an $O(\log n \log k)$
approximation for the \kv problem, and a bicriteria approximation
for the \bv\ problem that gives an $O(\frac{1}{\eps}\log^2 n)$
approximation, while violating the budget by a factor of at most
$3+\eps$.

Chandra Chekuri and Nitish Korula. Pruning 2-Connected Graphs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 119-130, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{chekuri_et_al:LIPIcs.FSTTCS.2008.1746, author = {Chekuri, Chandra and Korula, Nitish}, title = {{Pruning 2-Connected Graphs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {119--130}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1746}, URN = {urn:nbn:de:0030-drops-17469}, doi = {10.4230/LIPIcs.FSTTCS.2008.1746}, annote = {Keywords: 2-Connected Graphs, k-MST, Density, Approximation} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

We study single-sink network design problems in undirected graphs
with vertex connectivity requirements. The input to these problems
is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex
$r$, a set of terminals $T \subseteq V$, and integer $k$. The goal is
to connect each terminal $t \in T$ to $r$ via $k$ \emph{vertex-disjoint}
paths. In the {\em connectivity} problem, the objective is to find a
min-cost subgraph of $G$ that contains the desired paths. There is a
$2$-approximation for this problem when $k \le 2$ \cite{FleischerJW}
but for $k \ge 3$, the first non-trivial approximation was obtained
in the recent work of Chakraborty, Chuzhoy and Khanna
\cite{ChakCK08}; they describe and analyze an algorithm with an
approximation ratio of $O(k^{O(k^2)}\log^4 n)$ where $n=|V|$.
In this paper, inspired by the results and ideas in \cite{ChakCK08},
we show an $O(k^{O(k)}\log |T|)$-approximation bound for a simple
greedy algorithm. Our analysis is based on the dual of a natural
linear program and is of independent technical interest. We use the
insights from this analysis to obtain an $O(k^{O(k)}\log
|T|)$-approximation for the more general single-sink {\em
rent-or-buy} network design problem with vertex connectivity
requirements. We further extend the ideas to obtain a
poly-logarithmic approximation for the single-sink {\em buy-at-bulk}
problem when $k=2$ and the number of cable-types is a fixed
constant; we believe that this should extend to any fixed $k$. We
also show that for the non-uniform buy-at-bulk problem, for each
fixed $k$, a small variant of a simple algorithm suggested by
Charikar and Kargiazova \cite{CharikarK05} for the case of $k=1$
gives an $2^{O(\sqrt{\log |T|})}$ approximation for larger $k$.
These results show that for each of these problems, simple and
natural algorithms that have been developed for $k=1$ have good
performance for small $k > 1$.

Chandra Chekuri and Nitish Korula. Single-Sink Network Design with Vertex Connectivity Requirements. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 131-142, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{chekuri_et_al:LIPIcs.FSTTCS.2008.1747, author = {Chekuri, Chandra and Korula, Nitish}, title = {{Single-Sink Network Design with Vertex Connectivity Requirements}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {131--142}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1747}, URN = {urn:nbn:de:0030-drops-17475}, doi = {10.4230/LIPIcs.FSTTCS.2008.1747}, annote = {Keywords: Network Design, Vertex Connectivity, Buy-at-Bulk, Rent-or-Buy, Approximation} }

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