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The Maximum Binary Tree Problem

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Abstract

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 hard: it has no efficient \(\exp (-O(\log n/ \log \log n))\)-approximation 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 \({\mathbf {P}}\ne \mathbf {NP}\). 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 \(2^k \mathsf {poly}(n)\) 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.

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Acknowledgements

Karthekeyan Chandrasekaran is supported by NSF CCF-1814613 and NSF CCF-1907937. Elena Grigorescu, Young-San Lin, and Minshen Zhu are supported by NSF CCF-1910659 and NSF CCF-1910411. Gabriel Istrate was supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI project number PN-III-P4-ID-PCE-2016-0842, within PNCDI III. We thank our anonymous reviewers for many constructive comments that helped improve the presentation of the paper. A preliminary version of this work appeared in the proceedings of the 28th Annual European Symposium on Algorithms (ESA, 2020). This version contains complete proofs, an integer program, and an integrality gap instance.

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Authors and Affiliations

  1. University of Illinois, Urbana-Champaign, Urbana, USA

    Karthekeyan Chandrasekaran & Shubhang Kulkarni

  2. Purdue University, West Lafayette, USA

    Elena Grigorescu, Young-San Lin & Minshen Zhu

  3. West University of Timişoara, Romania and the e-Austria Research Institute, Timisoara, Romania

    Gabriel Istrate

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  1. Karthekeyan Chandrasekaran
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Correspondence to Karthekeyan Chandrasekaran.

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Chandrasekaran, K., Grigorescu, E., Istrate, G. et al. The Maximum Binary Tree Problem. Algorithmica 83, 2427–2468 (2021). https://doi.org/10.1007/s00453-021-00836-5

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