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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)

We consider an important generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset X ⊆ ℝ², partitioned into k color classes C₁, C₂, …, Cₖ ⊆ X. The goal is to find a minimum-cost Euclidean graph G such that every color class Cᵢ is connected in G. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to X. Each input point x ∈ X arrives with its color color(x) ∈ [k], and as usual for dynamic geometric streams, the input is restricted to the discrete grid {0, …, Δ}².
We design a single-pass streaming algorithm that uses poly(k ⋅ log Δ) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio α₂ (currently 1.1547 ≤ α₂ ≤ 1.214). This approximation guarantee matches the state of the art bound for streaming Steiner tree, i.e., when k = 1. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and has so far not been applied in the streaming setting.
We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires Ω(k) bits of space.

Artur Czumaj, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý. Streaming Algorithms for Geometric Steiner Forest. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{czumaj_et_al:LIPIcs.ICALP.2022.47, author = {Czumaj, Artur and Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Vesel\'{y}, Pavel}, title = {{Streaming Algorithms for Geometric Steiner Forest}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {47:1--47:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.47}, URN = {urn:nbn:de:0030-drops-163880}, doi = {10.4230/LIPIcs.ICALP.2022.47}, annote = {Keywords: Steiner forest, streaming, sublinear algorithms, dynamic programming} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known.
Our unified PTAS is based on the previous dynamic programming frameworks proposed in [Talwar STOC 2004] and [Bartal, Gottlieb, Krauthgamer STOC 2012]. However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

T-H. Hubert Chan, Haotian Jiang, and Shaofeng H.-C. Jiang. A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.ESA.2018.15, author = {Chan, T-H. Hubert and Jiang, Haotian and Jiang, Shaofeng H.-C.}, title = {{A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.15}, URN = {urn:nbn:de:0030-drops-94781}, doi = {10.4230/LIPIcs.ESA.2018.15}, annote = {Keywords: Doubling Dimension, Traveling Salesman Problem, Polynomial Time Approximation Scheme, Steiner Tree Problem, Prize Collecting} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We consider the online vector packing problem in which we have a d dimensional knapsack and items u with weight vectors w_u in R_+^d arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item u is accepted, w_u(i) units of capacity on dimension i will be taken up, for each i in [d]. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items S at the end of the algorithm, which is given by f(S) for some non-negative monotone submodular function f.
For any small constant epsilon > 0, we consider the special case that the weight of an item on every dimension is at most a (1- epsilon) fraction of the total capacity, and give a polynomial-time deterministic O(k / epsilon^2)-competitive algorithm for the problem, where k is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We first show that under the epsilon-slack assumption, no deterministic algorithm can obtain any o(k) competitive ratio, and no randomized algorithm can obtain any o(k / log k) competitive ratio. We then show that for the general case (when epsilon = 0), no randomized algorithm can obtain any o(k) competitive ratio.
In contrast to the (1+delta) competitive ratio achieved in Kesselheim et al. [STOC 2014] for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when |w_u|_infinity is arbitrarily small for all items u, it is impossible to achieve any o(log k / log log k) competitive ratio.

T.-H. Hubert Chan, Shaofeng H.-C. Jiang, Zhihao Gavin Tang, and Xiaowei Wu. Online Submodular Maximization Problem with Vector Packing Constraint. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan_et_al:LIPIcs.ESA.2017.24, author = {Chan, T.-H. Hubert and Jiang, Shaofeng H.-C. and Tang, Zhihao Gavin and Wu, Xiaowei}, title = {{Online Submodular Maximization Problem with Vector Packing Constraint}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.24}, URN = {urn:nbn:de:0030-drops-78190}, doi = {10.4230/LIPIcs.ESA.2017.24}, annote = {Keywords: Submodular Maximization, Free-disposal, Vector Packing} }

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