The Maximum Binary Tree Problem Young-San Lin Purdue University - - PowerPoint PPT Presentation

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 = (VT, ET) of G is a connected, acyclic subgraph

• f G where degT(v) ≤ 3 for each v ∈ VT.

Definitions

Definition (Binary Tree in Undirected Graph G)

A binary tree T = (VT, ET) of G is a connected, acyclic subgraph

• f G where degT(v) ≤ 3 for each v ∈ VT.

Definition (Binary Tree in Directed Graph G)

A binary tree T = (VT, ET) of G is a connected, acyclic subgraph

• f G where degout

T (v) ≤ 1 and degin T(v) ≤ 2 for each v ∈ VT.

Definitions

Definition (Binary Tree in Undirected Graph G)

A binary tree T = (VT, ET) of G is a connected, acyclic subgraph

• f G where degT(v) ≤ 3 for each v ∈ VT.

Definition (Binary Tree in Directed Graph G)

A binary tree T = (VT, ET) of G is a connected, acyclic subgraph

• f G where degout

T (v) ≤ 1 and degin T(v) ≤ 2 for each v ∈ VT.

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 degT(r) ≤ 2 (resp. degout

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 = (VT, ET) of G ◮ Goal: Maximize |VT|

Problems of Interest

Definition (UndirMBT, r-UndirMBT)

◮ Input: Undirected graph G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

Problems of Interest

Definition (UndirMBT, r-UndirMBT)

◮ Input: Undirected graph G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

Definition (DirMBT/DAGMBT,

)

◮ Input: Directed graph/DAG G = (V , E) ◮ Output: Binary Tree T = (VT, ET) of G ◮ Goal: Maximize |VT|

Problems of Interest

Definition (UndirMBT, r-UndirMBT)

◮ Input: Undirected graph G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

Definition (DirMBT/DAGMBT, r-DirMBT/r-DAGMBT)

◮ Input: Directed graph/DAG G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

Problems of Interest

Definition (UndirMBT, r-UndirMBT)

◮ Input: Undirected graph G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

Definition (DirMBT/DAGMBT, r-DirMBT/r-DAGMBT)

◮ Input: Directed graph/DAG G = (V , E) and root r ∈ V ◮ Output: Binary Tree T = (VT, ET) of G rooted at r ◮ Goal: Maximize |VT|

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 = σ1 · · · σn is said to be heapable if σ1 · · · σn 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 = σ1 · · · σn is said to be heapable if σ1 · · · σn 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 = σ1 · · · σn is said to be heapable if σ1 · · · σn 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 Gσ, s.t.

Motivation II — Connections to Sequence Heapability

Definition (Sequence Heapability)

A sequence σ = σ1 · · · σn is said to be heapable if σ1 · · · σn 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 Gσ, s.t.

• 1. Longest increasing subsequence of σ ⇐

⇒ longest path of Gσ.

Motivation II — Connections to Sequence Heapability

Definition (Sequence Heapability)

A sequence σ = σ1 · · · σn is said to be heapable if σ1 · · · σn 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 Gσ, s.t.

• 1. Longest increasing subsequence of σ ⇐

⇒ longest path of Gσ.

• 2. Longest heapable subsequence of σ ⇐

⇒ MBT of Gσ.

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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−ǫ n))-apx P = NP No poly-time Directed Ω

• 1

n1−ǫ

• apx

ETH Same as P = NP 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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−ǫ n))-apx P = NP Same as DAGs No poly-time Directed Ω

• 1

n1−ǫ

• apx

ETH Same as DAGs Same as P = NP 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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−ǫ n))-apx P = NP Same as DAGs No poly-time Directed Ω

• 1

n1−ǫ

• apx

ETH Same as DAGs Same as P = NP P = NP No poly-time Ω(1)-apx Undirected ETH No quasi-poly-time exp(−O(log1−ǫ n))-apx

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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−ǫ n))-apx P = NP Same as DAGs No poly-time Directed Ω

• 1

n1−ǫ

• apx

ETH Same as DAGs Same as P = NP P = NP No poly-time Ω(1)-apx No poly-time Ω(1)-apx Undirected ETH No quasi-poly-time No quasi-poly-time exp(−O(log0.63−ǫ n))-apx exp(−O(log1−ǫ n))-apx

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 DAGs ETH exp(−O(

log n log log n))-apx

Poly-time solvable No quasi-poly-time exp(−O(log1−ǫ n))-apx P = NP Same as DAGs No poly-time Directed Ω

• 1

n1−ǫ

• apx

ETH Same as DAGs Same as P = NP P = NP No poly-time Ω(1)-apx No poly-time Ω(1)-apx Undirected ETH No quasi-poly-time No quasi-poly-time exp(−O(log0.63−ǫ n))-apx exp(−O(log1−ǫ n))-apx

Our Algorithmic Result

Definition (k-BinaryTree)

◮ Input: A directed graph G = (V , E) and an integer k ◮ Goal: Accept if G contains a binary tree with k vertices; or reject if there is none

Our Algorithmic Result

Definition (k-BinaryTree)

◮ Input: A directed graph G = (V , E) and an integer k ◮ Goal: Accept if G contains a binary tree with k vertices; or reject if there is none

Remark

Search variant ≡p decision variant. Undirected ≤p directed.

Our Algorithmic Result

Definition (k-BinaryTree)

◮ Input: A directed graph G = (V , E) and an integer k ◮ Goal: Accept if G contains a binary tree with k vertices; or reject if there is none

Remark

Search variant ≡p decision variant. Undirected ≤p directed.

Theorem

There is a randomized algorithm for k-BinaryTree with

• ne-sided error and running time 2kpoly(n).

Overview

Introduction Inapproximability Results Hardness of DAGMBT k-BinaryTree via Multilinear Detection

Hardness of MBT

Theorem

There is no polynomial-time constant-factor approximation for DAGMBT and UndirMBT, unless P = NP.

Hardness of MBT

Theorem

There is no polynomial-time constant-factor approximation for DAGMBT and UndirMBT, unless P = NP.

Hardness of MBT

Theorem

There is no polynomial-time constant-factor approximation for DAGMBT and UndirMBT, unless P = NP.

Remark

DAGMBT ≡p r-DAGMBT.

Hardness of MBT

Theorem

There is no polynomial-time constant-factor approximation for r-DAGMBT and UndirMBT, unless P = NP.

Remark

DAGMBT ≡p r-DAGMBT.

Hardness of r-DAGMBT

The proof consists of two steps.

• 1. If there is a polynomial-time constant-factor approximation for

r-DAGMBT, then there is a PTAS for r-DAGMBT.

• 2. r-DAGMBT is APX-hard.

Hardness of r-DAGMBT

The proof consists of two steps.

• 1. If there is a polynomial-time constant-factor approximation for

r-DAGMBT, then there is a PTAS for r-DAGMBT.

• 2. r-DAGMBT is APX-hard.

This self-improving idea is due to Karger, Motwani and Ramkumar, whereby they proved similar hardness results for the Longest Path problem.

Hardness of r-DAGMBT

The proof consists of two steps.

• 1. If there is a polynomial-time constant-factor approximation for

r-DAGMBT, then there is a PTAS for r-DAGMBT.

• 2. r-DAGMBT is APX-hard.

This self-improving idea is due to Karger, Motwani and Ramkumar, whereby they proved similar hardness results for the Longest Path problem. The hardness for UndirMBT can be proved by a similar argument.

Hardness of r-DAGMBT

The proof consists of two steps.

• 1. If there is a polynomial-time constant-factor approximation for

r-DAGMBT, then there is a PTAS for r-DAGMBT.

• 2. r-DAGMBT is APX-hard.

This self-improving idea is due to Karger, Motwani and Ramkumar, whereby they proved similar hardness results for the Longest Path problem. The hardness for UndirMBT can be proved by a similar argument.

r-DAGMBT: constant-factor apx = ⇒ PTAS

Denote OPT(G) := #vertices in MBT of G. Suppose we have an α-approximation algorithm A for DAGMBT.

r-DAGMBT: constant-factor apx = ⇒ PTAS

Denote OPT(G) := #vertices in MBT of G. Suppose we have an α-approximation algorithm A for DAGMBT.

• 1. Run A on G 2 to get an α-approximate solution T2.

r-DAGMBT: constant-factor apx = ⇒ PTAS

Denote OPT(G) := #vertices in MBT of G. Suppose we have an α-approximation algorithm A for DAGMBT.

• 1. Run A on G 2 to get an α-approximate solution T2.
• 2. Use T2 to recover a solution T1 for G satisfying (roughly)

|V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

r-DAGMBT: constant-factor apx = ⇒ PTAS

Denote OPT(G) := #vertices in MBT of G. Suppose we have an α-approximation algorithm A for DAGMBT.

• 1. Run A on G 2 to get an α-approximate solution T2.
• 2. Use T2 to recover a solution T1 for G satisfying (roughly)

|V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

By squaring the graph roughly t =

• log2

log2 α log2(1−ε)

• times, we get a

(1 − ε)-approximation. The running time is polynomial in

• V
• G 2t
• .

Squared Graph Construction

Given G = (V , E) and root r G ′: add a source s, connect s to each vertex G 2: replace each vertex of V by G ′ G 2: connect the source and root according to E Set the root in the root copy as the root of G 2 r v1 v2 s G G ′ G 2 r v1 v2 → root of G 2

Squared Graph Construction

Given G = (V , E) and root r , define a squared graph G 2. G ′: add a source s, connect s to each vertex G 2: replace each vertex of V by G ′ G 2: connect the source and root according to E Set the root in the root copy as the root of G 2 r v1 v2 s G G ′ G 2 r v1 v2 → root of G 2

Squared Graph Construction

Given G = (V , E) and root r , define a squared graph G 2. G ′: add a source s, connect s to each vertex G 2: replace each vertex of V by G ′ G 2: connect the source and root according to E Set the root in the root copy as the root of G 2 r v1 v2 s G G ′ G 2 r v1 v2 → root of G 2

Squared Graph Construction

Given G = (V , E) and root r , define a squared graph G 2. G ′: add a source s, connect s to each vertex G 2: replace each vertex of V by G ′ G 2: connect the source and root according to E Set the root in the root copy as the root of G 2 r v1 v2 s G G ′ G 2 r v1 v2 → root of G 2

Squared Graph Construction

Given G = (V , E) and root r , define a squared graph G 2. G ′: add a source s, connect s to each vertex G 2: replace each vertex of V by G ′ G 2: connect the source and root according to E Set the root in the root copy as the root of G 2 r v1 v2 s G G ′ G 2 r v1 v2 → root of G 2

Self-improving Reduction

Use T2 to recover a solution T1 for G satisfying (roughly) |V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

Self-improving Reduction

Use T2 to recover a solution T1 for G satisfying (roughly) |V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

Lemma

OPT(G 2) ≥ OPT(G) · (OPT(G) + 1).

Self-improving Reduction

Use T2 to recover a solution T1 for G satisfying (roughly) |V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

Lemma

OPT(G 2) ≥ OPT(G) · (OPT(G) + 1). Proof sketch: Take any optimal solution T for G. We can find a copy of T 2 in G 2, which has size

• V
• T 2

≥ |V (T)| · (|V (T)| + 1) ≥ OPT(G) · (OPT(G) + 1) .

Self-improving Reduction

Use T2 to recover a solution T1 for G satisfying (roughly) |V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

Self-improving Reduction

Use T2 to recover a solution T1 for G satisfying (roughly) |V (T1)| ≥

• |V (T2)| ≥
• α · OPT (G 2) ≥ √α · OPT(G).

We can find a binary tree of size

• |V (T2)| in G by
• 1. assembling the vertex copies that participate in T2,
• 2. or restricting T2 to some vertex copy.

Hardness of r-DAGMBT

• 1. If there is a polynomial-time constant-factor approximation for

r-DAGMBT, then there is a PTAS for r-DAGMBT.

• 2. r-DAGMBT is APX-hard.

APX-hardness of r-DAGMBT

Reduction from Max-3-Colorable-Subgraph.

Definition (Max-3-Colorable-Subgraph)

◮ Input: An undirected graph G that is 3-colorable. ◮ Output: A 3-coloring of G that maximizes the fraction of properly colored edges.

APX-hardness of r-DAGMBT

Reduction from Max-3-Colorable-Subgraph.

Definition (Max-3-Colorable-Subgraph)

◮ Input: An undirected graph G that is 3-colorable. ◮ Output: A 3-coloring of G that maximizes the fraction of properly colored edges. 32

33 + ε

• approximation is NP-hard [AOW12’, GS13’].

Hardness of r-DAGMBT

With the self-improving reduction by squared graphs, we also have:

Hardness of r-DAGMBT

With the self-improving reduction by squared graphs, we also have:

• 1. If DAGMBT admits a polynomial-time

exp (−O (log n/ log log n))-approximation, then NP ⊆ DTIME

• exp
• O

√n

• , refuting ETH.

Hardness of r-DAGMBT

With the self-improving reduction by squared graphs, we also have:

• 1. If DAGMBT admits a polynomial-time

exp (−O (log n/ log log n))-approximation, then NP ⊆ DTIME

• exp
• O

√n

• , refuting ETH.
• 2. For any ε > 0, if DAGMBT admits a quasi-polynomial time

exp

• −O
• log1−ε n
• approximation, then

NP ⊆ DTIME

• exp
• logO(1/ε) n
• , thus refuting ETH.

Overview

Introduction Inapproximability Results Hardness of DAGMBT k-BinaryTree via Multilinear Detection

k-BinaryTree

Definition (k-BinaryTree)

◮ Input: A directed graph G = (V , E) and an integer k ◮ Goal: Accept if G contains a binary tree with k vertices; or reject if there is none

Remark

Search variant ≡p decision variant. Undirected ≤p directed.

Theorem

There is a randomized algorithm for k-BinaryTree with

• ne-sided error and running time 2kpoly(n).

Multi-variate Polynomials

Let F be a field. A degree-d polynomial f in m variables x1, x2, · · · , xm with coefficients in F has the form f (x1, x2, · · · , xm) =

• S : [m]→N≥0

|S|≤d

cS ·

• i∈[m]

xS(i)

i

, where cS ∈ F for all S.

Multilinear Terms in Polynomials

Definition (Multilinear terms)

A monomial p is multilinear if p has the form p = c ·

• i∈S

xi for some non-empty subset S ⊆ [m] and c ∈ F \ {0}.

Multilinear Terms in Polynomials

Definition (Multilinear terms)

A monomial p is multilinear if p has the form p = c ·

• i∈S

xi for some non-empty subset S ⊆ [m] and c ∈ F \ {0}. For example, the polynomial f (x1, x2, x3) = x2

1 + x1x2 + x2 2x3 + x1x2x3

has two multilinear terms x1x2 and x1x2x3.

Arithmetic Circuit

x1 x2 x3 + × + × f (x1, x2, x3) An arithmetic circuit C

• 1. The size of C is the number
• f gates in C.

2. f (x1, x2, x3) =(x1 + x2)(x1 + x2x3) =x2

1 + x1x2 + x2 2x3 + x1x2x3

is the sum-of-product expansion of C.

Multilinear Detection(Mld)

Definition (Mld)

◮ Input: A homogeneous polynomial f . ◮ Goal: Accept if f contains a multilinear term; reject

• therwise.

Multilinear Detection(Mld)

Definition (Mld)

◮ Input: A homogeneous polynomial f . ◮ Goal: Accept if f contains a multilinear term; reject

• therwise.

Remark

• 1. A polynomial is homogenous if every monomial has the same

degree.

• 2. f is represented as an arithmetic circuit of + and × gates

with no scalar multiplications.

• 3. If f is not homogeneous, we can homogenize it by introducing

new variables to the circuit.

Multilinear Detection(Mld)

Our algorithm relies on the following theorem by Ryan Williams.

Theorem (Williams 2009)

Let P(x1, · · · , xn) have degree at most k, represented by an arithmetic circuit of size s(n) with additive gates (of unbounded fan-in) and multiplicative gates (of fan-in two), and no scalar

• multiplications. There is a randomized algorithm that on every P

runs in 2kpoly(n) · s(n) time, outputs yes with high probability if there is a multilinear term in the sum-product expansion of P, and always outputs no if there is no multilinear term.

Reducing k-BinaryTree to Mld

Given a directed graph G and an integer k, we will construct a polynomial P(k)

G

with the following properties.

Reducing k-BinaryTree to Mld

Given a directed graph G and an integer k, we will construct a polynomial P(k)

G

with the following properties.

• 1. P(k)

G

is homogeneous and has degree k.

Reducing k-BinaryTree to Mld

Given a directed graph G and an integer k, we will construct a polynomial P(k)

G

with the following properties.

• 1. P(k)

G

is homogeneous and has degree k.

• 2. P(k)

G

can be represented as an arithmetic circuit of size poly(n).

Reducing k-BinaryTree to Mld

Given a directed graph G and an integer k, we will construct a polynomial P(k)

G

with the following properties.

• 1. P(k)

G

is homogeneous and has degree k.

• 2. P(k)

G

can be represented as an arithmetic circuit of size poly(n).

• 3. There is a one-to-one correspondence between binary trees of

size k in G and multilinear terms in P(k)

G .

A Warmup: from k-Path to Mld

Given G = (V , E), introduce variables x = {xv : v ∈ V }. The following construction P(k)

G (x) =

• (v1,··· ,vk) a path in G

k

• i=1

xvi would satisfy properties 1 and 3, but not 2!

A Warmup: from k-Path to Mld

Given G = (V , E), introduce variables x = {xv : v ∈ V }. The idea is to allow “self-intersections”: P(k)

G (x) =

• (v1,··· ,vk) a walk in G

k

• i=1

xvi would satisfy all 3 properties.

Multi-trees

We want to generalize the “self-intersection” idea to trees.

Multi-trees

We want to generalize the “self-intersection” idea to trees.

Definition (Multi-tree)

A multi-tree in a directed graph G is a pair T = (T, φ) where T is a directed tree and φ: V (T) → V (G) is a graph homomorphism from T to G.

Multi-trees

We want to generalize the “self-intersection” idea to trees.

Definition (Multi-tree)

A multi-tree in a directed graph G is a pair T = (T, φ) where T is a directed tree and φ: V (T) → V (G) is a graph homomorphism from T to G. ◮ We say T is rooted at v ∈ V (G) if φ(r) = v where r is the root of T. ◮ T is a binary multi-tree if T is a binary tree. ◮ When φ is injective we can find a copy of T in G.

Arithmetization of k-BinaryTree

Following the idea, let P(k)

G (x) =

• multi-tree T in G

|V (T)|=k

• u∈V (T)

xφ(u).

Arithmetization of k-BinaryTree

Following the idea, let P(k)

G (x) =

• multi-tree T in G

|V (T)|=k

• u∈V (T)

xφ(u). Properties 1 and 3 are satisfied. Property 2 is satisfied since there is a Dynamic Programming procedure for evaluating P(k)

G (x).

Arithmetization of k-BinaryTree

Fix a graph G = (V , E). For every integer k and v ∈ V define a polynomial P(k)

v

(x) =

• v-rooted T

|V (T)|=k

• u∈V (T)

xφ(u). Then P(k)

G (x) = v∈V P(k) v

(x).

Arithmetization of k-BinaryTree

Denote ∆in

v = {u : (u, v) ∈ E}.

◮ If k = 1, P(1)

v (x) = xv.

Arithmetization of k-BinaryTree

Denote ∆in

v = {u : (u, v) ∈ E}.

◮ If k = 1, P(1)

v (x) = xv.

◮ If ∆in

v = ∅, P(k) v

= xv · yk−1. (y is a dummy variable for homogenization purpose)

Arithmetization of k-BinaryTree

Denote ∆in

v = {u : (u, v) ∈ E}.

◮ If k = 1, P(1)

v (x) = xv.

◮ If ∆in

v = ∅, P(k) v

= xv · yk−1. (y is a dummy variable for homogenization purpose) ◮ In other cases, P(k)

v

= xv          

• u∈∆in

v

P(k−1)

u

• root(T) has
• nly 1 child

+

k−2

• ℓ=1

 

u1∈∆in

v

P(ℓ)

u1

 

• left subtree

has size ℓ

 

u2∈∆in

v

P(k−1−ℓ)

u2

 

• right subtree

has size k − 1 − ℓ

          .

Arithmetization of k-BinaryTree

Denote ∆in

v = {u : (u, v) ∈ E}.

◮ If k = 1, P(1)

v (x) = xv.

◮ If ∆in

v = ∅, P(k) v

= xv · yk−1. (y is a dummy variable for homogenization purpose) ◮ In other cases, P(k)

v

= xv          

• u∈∆in

v

P(k−1)

u

• root(T) has
• nly 1 child

+

k−2

• ℓ=1

 

u1∈∆in

v

P(ℓ)

u1

 

• left subtree

has size ℓ

 

u2∈∆in

v

P(k−1−ℓ)

u2

 

• right subtree

has size k − 1 − ℓ

          . The corresponding arithmetic circuit has size O(k2n).

Summary & Open Problems

◮ Assuming P = NP, no polynomial-time constant-factor approximations for DAGMBT and UndirMBT. (Open: Ω(1/n1−ε)-approximation for DAGMBT ?)

Summary & Open Problems

◮ Assuming P = NP, no polynomial-time constant-factor approximations for DAGMBT and UndirMBT. (Open: Ω(1/n1−ε)-approximation for DAGMBT ?) ◮ Assuming ETH, no quasi-polynomial-time exp(−O(log1−ε n))-approximation for DAGMBT; no quasi-polynomial-time exp(−O(log0.63−ε n))-approximation for UndirMBT. (Open: Bring exp(−O(log0.63−ε n)) to exp(−O(log1−ε n)) ?)

Summary & Open Problems

◮ Assuming P = NP, no polynomial-time constant-factor approximations for DAGMBT and UndirMBT. (Open: Ω(1/n1−ε)-approximation for DAGMBT ?) ◮ Assuming ETH, no quasi-polynomial-time exp(−O(log1−ε n))-approximation for DAGMBT; no quasi-polynomial-time exp(−O(log0.63−ε n))-approximation for UndirMBT. (Open: Bring exp(−O(log0.63−ε n)) to exp(−O(log1−ε n)) ?) ◮ There is a 2kpoly(n)-time randomized algorithm with

• ne-sided error for k-BinaryTree.

(Open: Improve running time to αkpoly(n) for α < 2 ?)

Summary & Open Problems

◮ Assuming P = NP, no polynomial-time constant-factor approximations for DAGMBT and UndirMBT. (Open: Ω(1/n1−ε)-approximation for DAGMBT ?) ◮ Assuming ETH, no quasi-polynomial-time exp(−O(log1−ε n))-approximation for DAGMBT; no quasi-polynomial-time exp(−O(log0.63−ε n))-approximation for UndirMBT. (Open: Bring exp(−O(log0.63−ε n)) to exp(−O(log1−ε n)) ?) ◮ There is a 2kpoly(n)-time randomized algorithm with

• ne-sided error for k-BinaryTree.

(Open: Improve running time to αkpoly(n) for α < 2 ?) Thanks!