The Maximum Binary Tree Problem Young-San Lin Purdue University August 25, 2020 joint work with Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni and Minshen Zhu
Overview Introduction Inapproximability Results Hardness of DAGMBT k - BinaryTree via Multilinear Detection
Overview Introduction Inapproximability Results Hardness of DAGMBT k - BinaryTree via Multilinear Detection
Introduction ◮ Degree-constrained subgraph problems: Find an optimal subgraph with specific properties satisfying degree constraints ◮ Minimum-cost Degree-Bounded Spanning Tree ◮ Minimum Subgraph of Minimum Degree d
Introduction ◮ Degree-constrained subgraph problems: Find an optimal subgraph with specific properties satisfying degree constraints ◮ Minimum-cost Degree-Bounded Spanning Tree ◮ Minimum Subgraph of Minimum Degree d ◮ Flip the objective and constraint ◮ The Minimum Degree Spanning Tree problem
Introduction ◮ Degree-constrained subgraph problems: Find an optimal subgraph with specific properties satisfying degree constraints ◮ Minimum-cost Degree-Bounded Spanning Tree ◮ Minimum Subgraph of Minimum Degree d ◮ Flip the objective and constraint ◮ The Minimum Degree Spanning Tree problem ◮ Have led to interesting techniques in approximation algorithms
Introduction ◮ Degree-constrained subgraph problems: Find an optimal subgraph with specific properties satisfying degree constraints ◮ Minimum-cost Degree-Bounded Spanning Tree ◮ Minimum Subgraph of Minimum Degree d ◮ Flip the objective and constraint ◮ The Minimum Degree Spanning Tree problem ◮ Have led to interesting techniques in approximation algorithms ◮ We are interested in finding maximum binary trees (MBT)
Definitions Definition (Binary Tree in Undirected Graph G ) A binary tree T = ( V T , E T ) of G is a connected, acyclic subgraph of G where deg T ( v ) ≤ 3 for each v ∈ V T .
Definitions Definition (Binary Tree in Undirected Graph G ) A binary tree T = ( V T , E T ) of G is a connected, acyclic subgraph of G where deg T ( v ) ≤ 3 for each v ∈ V T . Definition (Binary Tree in Directed Graph G ) A binary tree T = ( V T , E T ) of G is a connected, acyclic subgraph of G where deg out T ( v ) ≤ 1 and deg in T ( v ) ≤ 2 for each v ∈ V T .
Definitions Definition (Binary Tree in Undirected Graph G ) A binary tree T = ( V T , E T ) of G is a connected, acyclic subgraph of G where deg T ( v ) ≤ 3 for each v ∈ V T . Definition (Binary Tree in Directed Graph G ) A binary tree T = ( V T , E T ) of G is a connected, acyclic subgraph of G where deg out T ( v ) ≤ 1 and deg in T ( v ) ≤ 2 for each v ∈ V T . Definition (Rooted Binary Tree) A binary tree in an undirected (resp. directed) graph G = ( V , E ) is said to be rooted at r ∈ V if deg T ( r ) ≤ 2 (resp. deg out T ( r ) = 0).
A Directed Binary Tree r
A Directed Binary Tree r
A Directed Binary Tree r
Problems of Interest Definition ( UndirMBT , ) ◮ Input: Undirected graph G = ( V , E ) ◮ Output: Binary Tree T = ( V T , E T ) of G ◮ Goal: Maximize | V T |
Problems of Interest Definition ( UndirMBT , r-UndirMBT ) ◮ Input: Undirected graph G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T |
Problems of Interest Definition ( UndirMBT , r-UndirMBT ) ◮ Input: Undirected graph G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T | Definition ( DirMBT / DAGMBT , ) ◮ Input: Directed graph/DAG G = ( V , E ) ◮ Output: Binary Tree T = ( V T , E T ) of G ◮ Goal: Maximize | V T |
Problems of Interest Definition ( UndirMBT , r-UndirMBT ) ◮ Input: Undirected graph G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T | Definition ( DirMBT / DAGMBT , r-DirMBT / r-DAGMBT ) ◮ Input: Directed graph/DAG G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T |
Problems of Interest Definition ( UndirMBT , r-UndirMBT ) ◮ Input: Undirected graph G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T | Definition ( DirMBT / DAGMBT , r-DirMBT / r-DAGMBT ) ◮ Input: Directed graph/DAG G = ( V , E ) and root r ∈ V ◮ Output: Binary Tree T = ( V T , E T ) of G rooted at r ◮ Goal: Maximize | V T | Remark DAGMBT ≡ p r-DAGMBT . Unclear for general directed and undirected graphs.
Motivation I — Connections to Longest Path ◮ MBT can be viewed as a variant of the Longest Path problem
Motivation I — Connections to Longest Path ◮ MBT can be viewed as a variant of the Longest Path problem ◮ Longest Path reformulation: Find a maximum-sized tree in which every vertex has degree at most 2 for undirected graphs;
Motivation I — Connections to Longest Path ◮ MBT can be viewed as a variant of the Longest Path problem ◮ Longest Path reformulation: Find a maximum-sized tree in which every vertex has degree at most 2 for undirected graphs; in which every vertex has in and out degree at most 1 and out-degree of root is 0 for directed graphs
Motivation I — Connections to Longest Path ◮ MBT can be viewed as a variant of the Longest Path problem ◮ Longest Path reformulation: Find a maximum-sized tree in which every vertex has degree at most 2 for undirected graphs; in which every vertex has in and out degree at most 1 and out-degree of root is 0 for directed graphs ◮ MBT vs Longest Path illustration by picture:
Motivation II — Connections to Sequence Heapability Definition (Sequence Heapability) A sequence σ = σ 1 · · · σ n is said to be heapable if σ 1 · · · σ n can be inserted into a binary min-heap sequentially and in order. ◮ Introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO ′ 11)
Motivation II — Connections to Sequence Heapability Definition (Sequence Heapability) A sequence σ = σ 1 · · · σ n is said to be heapable if σ 1 · · · σ n can be inserted into a binary min-heap sequentially and in order. ◮ Introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO ′ 11) ◮ This problem can be viewed as a variant of the longest increasing subsequence.
Motivation II — Connections to Sequence Heapability Definition (Sequence Heapability) A sequence σ = σ 1 · · · σ n is said to be heapable if σ 1 · · · σ n can be inserted into a binary min-heap sequentially and in order. ◮ Introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO ′ 11) ◮ This problem can be viewed as a variant of the longest increasing subsequence. ◮ Given σ , we can construct a DAG G σ , s.t.
Motivation II — Connections to Sequence Heapability Definition (Sequence Heapability) A sequence σ = σ 1 · · · σ n is said to be heapable if σ 1 · · · σ n can be inserted into a binary min-heap sequentially and in order. ◮ Introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO ′ 11) ◮ This problem can be viewed as a variant of the longest increasing subsequence. ◮ Given σ , we can construct a DAG G σ , s.t. 1. Longest increasing subsequence of σ ⇐ ⇒ longest path of G σ .
Motivation II — Connections to Sequence Heapability Definition (Sequence Heapability) A sequence σ = σ 1 · · · σ n is said to be heapable if σ 1 · · · σ n can be inserted into a binary min-heap sequentially and in order. ◮ Introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO ′ 11) ◮ This problem can be viewed as a variant of the longest increasing subsequence. ◮ Given σ , we can construct a DAG G σ , s.t. 1. Longest increasing subsequence of σ ⇐ ⇒ longest path of G σ . 2. Longest heapable subsequence of σ ⇐ ⇒ MBT of G σ .
Our Hardness Results APX is α -approximation if APX ≥ α · OPT, where α ∈ (0 , 1). Family Assumption Max Binary Tree Longest Path P � = NP DAGs ETH P � = NP Directed ETH P � = NP Undirected ETH
Our Hardness Results APX is α -approximation if APX ≥ α · OPT, where α ∈ (0 , 1). Family Assumption Max Binary Tree Longest Path P � = NP Poly-time solvable DAGs ETH Poly-time solvable P � = NP Directed ETH P � = NP Undirected ETH
Our Hardness Results APX is α -approximation if APX ≥ α · OPT, where α ∈ (0 , 1). Family Assumption Max Binary Tree Longest Path P � = NP No poly-time Ω(1)-apx Poly-time solvable No poly-time log n DAGs ETH exp( − O ( log log n ))-apx Poly-time solvable No quasi-poly-time exp( − O (log 1 − ǫ n ))-apx P � = NP Directed ETH P � = NP Undirected ETH
Our Hardness Results APX is α -approximation if APX ≥ α · OPT, where α ∈ (0 , 1). Family Assumption Max Binary Tree Longest Path P � = NP No poly-time Ω(1)-apx Poly-time solvable No poly-time log n DAGs ETH exp( − O ( log log n ))-apx Poly-time solvable No quasi-poly-time exp( − O (log 1 − ǫ n ))-apx P � = NP No poly-time � 1 � Directed Ω -apx n 1 − ǫ ETH Same as P � = NP P � = NP Undirected ETH
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