TY - JOUR

T1 - The Maximum Binary Tree Problem

AU - Chandrasekaran, Karthekeyan

AU - Grigorescu, Elena

AU - Istrate, Gabriel

AU - Kulkarni, Shubhang

AU - Lin, Young San

AU - Zhu, Minshen

N1 - Funding Information:
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.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - 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 0.63n)) -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 kpoly(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.

AB - 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 0.63n)) -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 kpoly(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.

KW - Fixed-parameter tractability

KW - Inapproximability

KW - Maximum binary tree

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U2 - 10.1007/s00453-021-00836-5

DO - 10.1007/s00453-021-00836-5

M3 - Article

AN - SCOPUS:85107384065

VL - 83

SP - 2427

EP - 2468

JO - Algorithmica (New York)

JF - Algorithmica (New York)

SN - 0178-4617

IS - 8

ER -