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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

In the k-outconnected directed Steiner tree problem (k-DST), we are given an n-vertex directed graph G = (V,E) with edge costs, a connectivity requirement k, a root r ∈ V and a set of terminals T ⊆ V. The goal is to find a minimum-cost subgraph H ⊆ G that has k edge-disjoint paths from the root vertex r to every terminal t ∈ T. The problem is NP-hard, and inapproximability results are known in several parameters, e.g., hardness in terms of n: log^{2-ε}n-hardness for k = 1 [Halperin and Krauthgamer, STOC'03], 2^{log^{1-ε}n}-hardness for general case [Cheriyan, Laekhanukit, Naves and Vetta, SODA'12], hardness in terms of k [Cheriyan et al., SODA'12; Laekhanukit, SODA'14; Manurangsi, IPL'19] and hardness in terms of |T| [Laekhanukit, SODA'14].
In this paper, we show the approximation hardness of k-DST for various parameters.
- Ω(|T|/log |T|)-approximation hardness, which holds under the standard complexity assumption NP≠ ZPP. The inapproximability ratio is tightened to Ω(|T|) under the Strongish Planted Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021]. The latter hardness result matches the approximation ratio of |T| obtained by a trivial approximation algorithm, thus closing the long-standing open problem.
- Ω(2^{k/2} / k)-approximation hardness for the general case of k-DST under the assumption NP≠ZPP. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in k.
- Ω((k/L)^{L/4})-approximation hardness for k-DST on L-layered graphs for L ≤ O(log n). This almost matches the approximation ratio of O(k^{L-1}⋅ L ⋅ log |T|) achieved in O(n^L)-time due to Laekhanukit [ICALP'16].
We further extend our hardness results in terms of |T| to the undirected cases of k-DST, namely the single-source k-vertex-connected Steiner tree and the k-edge-connected group Steiner tree problems. Thus, we obtain Ω(|T|/log |T|) and Ω(|T|) approximation hardness for both problems under the assumption NP≠ ZPP and the Strongish Planted Clique Hypothesis, respectively. This again matches the upper bound obtained by trivial algorithms.

Chao Liao, Qingyun Chen, Bundit Laekhanukit, and Yuhao Zhang. Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{liao_et_al:LIPIcs.ICALP.2022.89, author = {Liao, Chao and Chen, Qingyun and Laekhanukit, Bundit and Zhang, Yuhao}, title = {{Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {89:1--89:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.89}, URN = {urn:nbn:de:0030-drops-164309}, doi = {10.4230/LIPIcs.ICALP.2022.89}, annote = {Keywords: Directed Steiner Tree, Hardness of Approximation, Fault-Tolerant and Survivable Network Design} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in ℝ^d, with d ≥ 3, are NP-hardness and an O(log³ n)-approximation algorithm which is based on a reduction to the group Steiner tree problem.
We show that TSP with lines in ℝ^d is APX-hard for any d ≥ 3. More generally, this implies that TSP with k-dimensional flats does not admit a PTAS for any 1 ≤ k ≤ d-2 unless P = NP, which gives a complete classification regarding the existence of polynomial time approximation schemes for these problems, as there are known PTASes for k = 0 (i.e., points) and k = d-1 (hyperplanes). We are able to give a stronger inapproximability factor for d = O(log n) by showing that TSP with lines does not admit a (2-ε)-approximation in d dimensions under the Unique Games Conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an O(log² n)-approximation algorithm for the problem, albeit with a running time of n^{O(log log n)}.

Antonios Antoniadis, Sándor Kisfaludi-Bak, Bundit Laekhanukit, and Daniel Vaz. On the Approximability of the Traveling Salesman Problem with Line Neighborhoods. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{antoniadis_et_al:LIPIcs.SWAT.2022.10, author = {Antoniadis, Antonios and Kisfaludi-Bak, S\'{a}ndor and Laekhanukit, Bundit and Vaz, Daniel}, title = {{On the Approximability of the Traveling Salesman Problem with Line Neighborhoods}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {10:1--10:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.10}, URN = {urn:nbn:de:0030-drops-161706}, doi = {10.4230/LIPIcs.SWAT.2022.10}, annote = {Keywords: Traveling Salesman with neighborhoods, Group Steiner Tree, Geometric approximation algorithms} }

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APPROX

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

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/log log k)-approximation algorithms for the problem, which are tight under popular complexity assumptions.
In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasi-polynomial time (O(log n log k), O(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem.
While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.

Xiangyu Guo, Guy Kortsarz, Bundit Laekhanukit, Shi Li, Daniel Vaz, and Jiayi Xian. On Approximating Degree-Bounded Network Design Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2020.39, author = {Guo, Xiangyu and Kortsarz, Guy and Laekhanukit, Bundit and Li, Shi and Vaz, Daniel and Xian, Jiayi}, title = {{On Approximating Degree-Bounded Network Design Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {39:1--39:21}, 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.2020.39}, URN = {urn:nbn:de:0030-drops-126420}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.39}, annote = {Keywords: Directed Steiner Tree, Group Steiner Tree, degree-bounded} }

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APPROX

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

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ≥ 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ≥ 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem.

Chun-Hsiang Chan, Bundit Laekhanukit, Hao-Ting Wei, and Yuhao Zhang. Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 63:1-63:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.APPROX/RANDOM.2020.63, author = {Chan, Chun-Hsiang and Laekhanukit, Bundit and Wei, Hao-Ting and Zhang, Yuhao}, title = {{Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {63:1--63:20}, 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.2020.63}, URN = {urn:nbn:de:0030-drops-126667}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.63}, annote = {Keywords: Approximation Algorithms, Network Design, Directed Graphs} }

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

In the Group Steiner Tree problem (GST), we are given a (edge or vertex)-weighted graph G=(V,E) on n vertices, together with a root vertex r and a collection of groups {S_i}_{i in [h]}: S_i subseteq V(G). The goal is to find a minimum-cost subgraph H that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group S_i has a demand k_i in [k], k in N, and we wish to find a minimum-cost subgraph H subseteq G such that, for each group S_i, there is a vertex in the group that is connected to the root via k_i (vertex or edge) disjoint paths.
While GST admits O(log^2 n log h) approximation, its higher connectivity variants are known to be Label-Cover hard, and for the vertex-weighted version, the hardness holds even when k=2 (it is widely believed that there is no subpolynomial approximation for the Label-Cover problem [Bellare et al., STOC 1993]). More precisely, the problem admits no 2^{log^{1-epsilon}n}-approximation unless NP subseteq DTIME(n^{polylog(n)}). Previously, positive results were known only for the edge-weighted version when k=2 [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where k_i disjoint paths from r may end at different vertices in a group [Chalermsook et al., SODA 2015], for which the authors gave a bicriteria approximation. For k >= 3, there is no non-trivial approximation algorithm known for edge-weighted Restricted Group SNDP, except for the special case of the relaxed variant on trees (folklore).
Our main result is an O(log n log h) approximation algorithm for Restricted Group SNDP that runs in time n^{f(k, w)}, where w is the treewidth of the input graph. Our algorithm works for both edge and vertex weighted variants, and the approximation ratio nearly matches the lower bound when k and w are constants. The key to achieving this result is a non-trivial extension of a framework introduced in [Chalermsook et al., SODA 2017]. This framework first embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.

Parinya Chalermsook, Syamantak Das, Guy Even, Bundit Laekhanukit, and Daniel Vaz. Survivable Network Design for Group Connectivity in Low-Treewidth Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chalermsook_et_al:LIPIcs.APPROX-RANDOM.2018.8, author = {Chalermsook, Parinya and Das, Syamantak and Even, Guy and Laekhanukit, Bundit and Vaz, Daniel}, title = {{Survivable Network Design for Group Connectivity in Low-Treewidth Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {8:1--8:19}, 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.2018.8}, URN = {urn:nbn:de:0030-drops-94127}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.8}, annote = {Keywords: Approximation Algorithms, Hardness of Approximation, Survivable Network Design, Group Steiner Tree} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Every graph G can be represented by a collection of equi-radii spheres in a d-dimensional metric Delta such that there is an edge uv in G if and only if the spheres corresponding to u and v intersect. The smallest integer d such that G can be represented by a collection of spheres (all of the same radius) in Delta is called the sphericity of G, and if the collection of spheres are non-overlapping, then the value d is called the contact-dimension of G. In this paper, we study the sphericity and contact dimension of the complete bipartite graph K_{n,n} in various L^p-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.

Roee David, Karthik C. S., and Bundit Laekhanukit. On the Complexity of Closest Pair via Polar-Pair of Point-Sets. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{david_et_al:LIPIcs.SoCG.2018.28, author = {David, Roee and C. S., Karthik and Laekhanukit, Bundit}, title = {{On the Complexity of Closest Pair via Polar-Pair of Point-Sets}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.28}, URN = {urn:nbn:de:0030-drops-87412}, doi = {10.4230/LIPIcs.SoCG.2018.28}, annote = {Keywords: Contact dimension, Sphericity, Closest Pair, Fine-Grained Complexity} }

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

Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H subseteq G that connects r to each terminal t in T by k edge-disjoint r, t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k=1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14], by Laekhanukit [SODA'14] and in a survey by Kortsarz and Nutov [Handbook of Approximation Algorithms and Metaheuristics], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in a special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k=2 and h=2.
In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D*k^{D-1}*log(n))-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r, t-paths, each of length at most D, for every terminal t in T. For the case k=1 (DST), our algorithm yields an approximation ratio of O(D*log(h)), thus implying an O(log^3(h))-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Our algorithm is based on an LP-formulation that allows us to embed a solution to a tree-instance of GST, which does not preserve connectivity. We show, however, that one can randomly extract a solution of k-DST from the tree-instance of GST.
Our algorithm is almost tight when k and D are constants since the case that k=1 and D=3 is NP-hard to approximate to within a factor of O(log(h)), and our algorithm archives the same approximation ratio for this special case. We also remark that the k^{1/4-epsilon}-hardness instance of k-DST is a DAG with 6 layers, and our algorithm gives O(k^5*log(n))-approximation for this special case. Consequently, as our algorithm works for general graphs, we obtain an O(D*k^{D-1}*log(n))-approximation algorithm for a D-shallow instance of the k edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k edgedisjoint paths.

Bundit Laekhanukit. Approximating Directed Steiner Problems via Tree Embedding. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 74:1-74:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{laekhanukit:LIPIcs.ICALP.2016.74, author = {Laekhanukit, Bundit}, title = {{Approximating Directed Steiner Problems via Tree Embedding}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {74:1--74:13}, 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.2016.74}, URN = {urn:nbn:de:0030-drops-62100}, doi = {10.4230/LIPIcs.ICALP.2016.74}, annote = {Keywords: Approximation Algorithms, Network Design, Graph Connectivity, Directed Graph} }

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