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APPROX

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

In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied.
We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting:
- An Õ(n^{4/5 + ε})-approximation algorithm for the weighted pairwise spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n^{3/5 + ε})-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020).
- An Õ(n^{1/2+ε})-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives an Õ(n^{1/2})-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. Our results include the following:
- An Õ(k^{1/2 + ε})-competitive algorithm for the online weighted pairwise spanner problem. The state-of-the-art results are an Õ(n^{4/5})-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k^{1/2 + ε}), Õ(n^{2/3 + ε})}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021).
- An Õ(k^ε)-competitive algorithm for the online weighted single-source (or single-sink) spanner problem. Without distance constraints, this problem is equivalent to the online directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is an Õ(k^ε)-competitive algorithm, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018). Our online results also imply efficient approximation algorithms for the corresponding offline problems. To the best of our knowledge, these are the first approximation (online) polynomial-time algorithms with sublinear approximation (competitive) ratios for the weighted spanner problems.

Elena Grigorescu, Nithish Kumar, and Young-San Lin. Approximation Algorithms for Directed Weighted Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2023.8, author = {Grigorescu, Elena and Kumar, Nithish and Lin, Young-San}, title = {{Approximation Algorithms for Directed Weighted Spanners}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {8:1--8:23}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.8}, URN = {urn:nbn:de:0030-drops-188335}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.8}, annote = {Keywords: directed weighted spanners, linear programming, junction tree} }

Document

APPROX

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

We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following:
- For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020).
- For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018).
To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.

Elena Grigorescu, Young-San Lin, and Kent Quanrud. Online Directed Spanners and Steiner Forests. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 5:1-5:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2021.5, author = {Grigorescu, Elena and Lin, Young-San and Quanrud, Kent}, title = {{Online Directed Spanners and Steiner Forests}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {5:1--5:25}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.5}, URN = {urn:nbn:de:0030-drops-146987}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.5}, annote = {Keywords: online directed pairwise spanners, online directed Steiner forests, online covering/packing linear programming, primal-dual approach} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence and maximum-sized binary tree. We introduce alphabet size as a new parameter in the study of computational problems in permutation DAGs and show that this parameter with respect to a fixed topological ordering admits a complete characterization and a polynomial time algorithm. We believe that this parameter is likely to be useful in the context of optimization problems defined over permutation DAGs.

Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu. Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chandrasekaran_et_al:LIPIcs.IPEC.2020.7, author = {Chandrasekaran, Karthekeyan and Grigorescu, Elena and Istrate, Gabriel and Kulkarni, Shubhang and Lin, Young-San and Zhu, Minshen}, title = {{Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.7}, URN = {urn:nbn:de:0030-drops-133102}, doi = {10.4230/LIPIcs.IPEC.2020.7}, annote = {Keywords: maximum binary tree, heapability, permutation directed acyclic graphs} }

Document

**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph.
The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ≠ NP.
In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.

Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu. The Maximum Binary Tree Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chandrasekaran_et_al:LIPIcs.ESA.2020.30, author = {Chandrasekaran, Karthekeyan and Grigorescu, Elena and Istrate, Gabriel and Kulkarni, Shubhang and Lin, Young-San and Zhu, Minshen}, title = {{The Maximum Binary Tree Problem}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {30:1--30:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.30}, URN = {urn:nbn:de:0030-drops-128967}, doi = {10.4230/LIPIcs.ESA.2020.30}, annote = {Keywords: maximum binary tree, heapability, inapproximability, fixed-parameter tractability} }

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