• DOI:10.1007/s00453-021-00836-5
• Corpus ID: 202583354

The Maximum Binary Tree Problem

@article{Chandrasekaran2019TheMB,
  title={The Maximum Binary Tree Problem},
  author={Karthekeyan Chandrasekaran and Elena Grigorescu and Gabriel Istrate and Shubhang Kulkarni and Young-San Lin and Minshen Zhu},
  journal={Algorithmica},
  year={2019},
  volume={83},
  pages={2427 - 2468}
}

• Karthekeyan Chandrasekaran, Elena Grigorescu, +3 authors Minshen Zhu
• Published 17 September 2019
• Computer Science, Mathematics
• Algorithmica

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(logn/loglogn))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (-O(\log n/ \log \log n))$$\end{document}-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(log0.63n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (-O(\log ^{0.63}{n}))$$\end{document}-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {P}}\ne \mathbf {NP}$$\end{document}. 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 2kpoly(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^k \mathsf {poly}(n)$$\end{document} 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.