Finding Triangles for Maximum Planar Subgraphs Ghent Graph Theory - - PowerPoint PPT Presentation

Finding Triangles for Maximum Planar Subgraphs

Ghent Graph Theory Workshop 2017

Parinya Chalermsooka Andreas Schmidb

Ghent, August 18, 2017 aAalto University, Espoo, Finland bMax Planck Institute for Informatics, Saarbrücken, Germany

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum.

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum. NP-hard,

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum. NP-hard, APX-hard.

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum. NP-hard, APX-hard. Trivial: 1

3-approximation.

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum. NP-hard, APX-hard. Trivial: 1

3-approximation.

Best known: 4

9-approximation. [C˘

alinescu et al. SODA 96]

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

The Problem

Definition (Maximum Planar Subgraph Problem (MPS))

Given a graph G = (V, E) find a planar subgraph H = (V, E′) where E′ ⊆ E is maximum. NP-hard, APX-hard. Trivial: 1

3-approximation.

Best known: 4

9-approximation. [C˘

alinescu et al. SODA 96] Many applications: Circuit design, factory layout, graph drawing problems.

Ghent, August 18, 2017 2/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Related Results

2 3-approximation for Maximum Outerplanar Subgraph Problem.

Ghent, August 18, 2017 3/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Related Results

2 3-approximation for Maximum Outerplanar Subgraph Problem.

( 1

3 + 1 72)-approximation for weighted MPS.[C˘

alinescu et al. 2003]

Ghent, August 18, 2017 3/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Related Results

2 3-approximation for Maximum Outerplanar Subgraph Problem.

( 1

3 + 1 72)-approximation for weighted MPS.[C˘

alinescu et al. 2003] MPS remains NP-hard in cubic graphs.[Faria et al. 2004]

Ghent, August 18, 2017 3/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Augment a spanning tree by forming edge-disjoint triangles

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure This gives a

7 18-approximation

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure This gives a

7 18-approximation

A maximum triangular structure gives a 4

9-approximation

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

How to beat 1/3 [C˘ alinescu et al. 96]

Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure This gives a

7 18-approximation

A maximum triangular structure gives a 4

9-approximation

Uses linear matroid parity algorithm as a black box.

Ghent, August 18, 2017 4/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G.

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G.

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G. Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G. Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E′ := E′ ∪ E(t).

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G. Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E′ := E′ ∪ E(t). Repeat until there are no more triangles in G.

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G. Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E′ := E′ ∪ E(t). Repeat until there are no more triangles in G.

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Start a new component C from a single triangle in G. Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E′ := E′ ∪ E(t). Repeat until there are no more triangles in G.

Ghent, August 18, 2017 5/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Works great in practice.

Ghent, August 18, 2017 6/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Works great in practice. Runs in linear time in bounded degree graphs.

Ghent, August 18, 2017 6/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Works great in practice. Runs in linear time in bounded degree graphs. Proven to give a

7 18-approximation.

Ghent, August 18, 2017 6/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Candidate Algorithm (08)

Works great in practice. Runs in linear time in bounded degree graphs. Proven to give a

7 18-approximation.

Conjectured to give a

4 9-approximation.

Ghent, August 18, 2017 6/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Point of This Work

Is there hope to reach the 4

9-approximation by using non

edge-disjoint triangles?

Ghent, August 18, 2017 7/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Point of This Work

Is there hope to reach the 4

9-approximation by using non

edge-disjoint triangles? ⇒ Start by improving over the best known greedy algorithm!

Ghent, August 18, 2017 7/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Focusing on Triangles

Ghent, August 18, 2017 8/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Focusing on Triangles

We introduce a new problem that captures the previous ideas.

Ghent, August 18, 2017 8/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Focusing on Triangles

We introduce a new problem that captures the previous ideas.

Definition (Maximum Planar Triangles Problem (MPT))

Given a graph G = (V, E) find a plane subgraph H = (V, E′) with a maximum number of triangular faces.

Ghent, August 18, 2017 8/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Focusing on Triangles

We introduce a new problem that captures the previous ideas.

Definition (Maximum Planar Triangles Problem (MPT))

Given a graph G = (V, E) find a plane subgraph H = (V, E′) with a maximum number of triangular faces. We can show that:

Ghent, August 18, 2017 8/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Focusing on Triangles

We introduce a new problem that captures the previous ideas.

Definition (Maximum Planar Triangles Problem (MPT))

Given a graph G = (V, E) find a plane subgraph H = (V, E′) with a maximum number of triangular faces. We can show that:

Theorem

MPT is NP-hard. Via a reduction from Hamiltonian path in bipartite graphs.

Ghent, August 18, 2017 8/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

From MPT to MPS

We quantify the connection between MPS and MPT.

Ghent, August 18, 2017 9/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

From MPT to MPS

We quantify the connection between MPS and MPT.

Theorem

If there is a β-approximation algorithm for MPT, then there is a min( 1

2, 1 3 + 2β 3 )-approximation algorithm for MPS.

Ghent, August 18, 2017 9/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

From MPT to MPS

We quantify the connection between MPS and MPT.

Theorem

If there is a β-approximation algorithm for MPT, then there is a min( 1

2, 1 3 + 2β 3 )-approximation algorithm for MPS.

( 1

6 + ε) for MPT ⇒ Improvement in MPS.

Ghent, August 18, 2017 9/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Ghent, August 18, 2017 10/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

All previous greedy algorithms fit in our framework.

Ghent, August 18, 2017 10/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

All previous greedy algorithms fit in our framework. We think analyzing MPT is cleaner than MPS.

Ghent, August 18, 2017 10/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

All previous greedy algorithms fit in our framework. We think analyzing MPT is cleaner than MPS. We settled Poranen’s Conjecture when analyzing his algorithm in our framework for MPT.

Ghent, August 18, 2017 10/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

All previous greedy algorithms fit in our framework. We think analyzing MPT is cleaner than MPS. We settled Poranen’s Conjecture when analyzing his algorithm in our framework for MPT.

Theorem

Growing components from triangles gives at most a

7 18-approximation for MPS.

Ghent, August 18, 2017 10/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Let C be the set of connected components in G′ = (V, E′).

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Let C be the set of connected components in G′ = (V, E′).

Definition

An (H, P)-rule is a pair where H is a graph (pattern) P = (V1, V2, . . . , Vk) is a partition of V(H).

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Let C be the set of connected components in G′ = (V, E′).

Definition

An (H, P)-rule is a pair where H is a graph (pattern) P = (V1, V2, . . . , Vk) is a partition of V(H). An (H, P)-rule applies to G′ if:

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Let C be the set of connected components in G′ = (V, E′).

Definition

An (H, P)-rule is a pair where H is a graph (pattern) P = (V1, V2, . . . , Vk) is a partition of V(H). An (H, P)-rule applies to G′ if: there is a subgraph H′ in G that is isomorphic to H and

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Match and Merge for MPT

Let C be the set of connected components in G′ = (V, E′).

Definition

An (H, P)-rule is a pair where H is a graph (pattern) P = (V1, V2, . . . , Vk) is a partition of V(H). An (H, P)-rule applies to G′ if: there is a subgraph H′ in G that is isomorphic to H and components in H′ induced by C match those in H induced by P.

Ghent, August 18, 2017 11/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Edge-Disjoint Triangles

Ghent, August 18, 2017 12/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Edge-Disjoint Triangles

Can be expressed by a (K3, ({u}, {v}, {w}))-rule (or K3-rule).

Ghent, August 18, 2017 12/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Edge-Disjoint Triangles

Can be expressed by a (K3, ({u}, {v}, {w}))-rule (or K3-rule). Algorithm:

• 1. Check if K3-rule applies.
• 2. While |E′| increases go back to (1).

u v w

Ghent, August 18, 2017 12/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Edge-Disjoint Triangles

Can be expressed by a (K3, ({u}, {v}, {w}))-rule (or K3-rule). Algorithm:

• 1. Check if K3-rule applies.
• 2. While |E′| increases go back to (1).

u v w

Theorem

Collecting edge-disjoint triangles gives a

1 12-approximation for

MPT.

Ghent, August 18, 2017 12/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Algorithm

Algorithm:

• 1. Check if (K3, ({u, v}, {w}))-rule applies
• 2. If not, check if K3-rule applies.
• 3. While |E′| increases go back to (1).

v u w

Ghent, August 18, 2017 13/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Poranen’s Algorithm

Algorithm:

• 1. Check if (K3, ({u, v}, {w}))-rule applies
• 2. If not, check if K3-rule applies.
• 3. While |E′| increases go back to (1).

v u w

Theorem

Growing components from triangles gives a

1 12-approximation for

MPT.

Ghent, August 18, 2017 13/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Insights From Analysis in M&M

Both algorithms share the same weakness. We compare a component C ∈ C to the subgraph induced by V(C) in OPTMPT.

Ghent, August 18, 2017 14/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Insights From Analysis in M&M

Both algorithms share the same weakness. We compare a component C ∈ C to the subgraph induced by V(C) in OPTMPT. Locally C is much better than a

1 12-approximation.

Ghent, August 18, 2017 14/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Insights From Analysis in M&M

Both algorithms share the same weakness. We compare a component C ∈ C to the subgraph induced by V(C) in OPTMPT. Locally C is much better than a

1 12-approximation.

C

Ghent, August 18, 2017 14/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Insights From Analysis in M&M

Both algorithms share the same weakness. We compare a component C ∈ C to the subgraph induced by V(C) in OPTMPT. Locally C is much better than a

1 12-approximation.

C But OPTMPT can have many triangles like this:

Ghent, August 18, 2017 14/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Insights From Analysis in M&M

Both algorithms share the same weakness. We compare a component C ∈ C to the subgraph induced by V(C) in OPTMPT. Locally C is much better than a

1 12-approximation.

C But OPTMPT can have many triangles like this:

C C

Ghent, August 18, 2017 14/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

Ghent, August 18, 2017 15/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

Let D be a diamond graph (i.e. K4 with one edge removed).

Ghent, August 18, 2017 15/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

Let D be a diamond graph (i.e. K4 with one edge removed). Algorithm:

• 1. Check if D-rule applies
• 2. While |E′| increases go back to

(1).

• 3. Check if K3-rule applies.
• 4. While |E′| increases go back to

(3).

Ghent, August 18, 2017 15/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

We show that this breaks

1 12 for MPT.

Ghent, August 18, 2017 16/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

We show that this breaks

1 12 for MPT.

Theorem

Collecting diamonds and triangles gives a

1 11-approximation for

MPT.

Ghent, August 18, 2017 16/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

We show that this breaks

1 12 for MPT.

Theorem

Collecting diamonds and triangles gives a

1 11-approximation for

MPT. Leading to the first greedy algorithm that breaks

7 18 for MPS.

Ghent, August 18, 2017 16/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

A New Greedy Algorithm

We show that this breaks

1 12 for MPT.

Theorem

Collecting diamonds and triangles gives a

1 11-approximation for

MPT. Leading to the first greedy algorithm that breaks

7 18 for MPS.

Corollary

Collecting diamonds and triangles gives a 13

33-approximation for

MPS.

Ghent, August 18, 2017 16/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Open Problems

Hardness of approximating MPT?

Ghent, August 18, 2017 17/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Open Problems

Hardness of approximating MPT? Do diamonds give a 1

9-approximation for MPT?

Ghent, August 18, 2017 17/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Open Problems

Hardness of approximating MPT? Do diamonds give a 1

9-approximation for MPT?

Is there a rule that gives a ( 1

6 + ε)-approximation for MPT?

Ghent, August 18, 2017 17/17

Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems

Open Problems

Hardness of approximating MPT? Do diamonds give a 1

9-approximation for MPT?

Is there a rule that gives a ( 1

6 + ε)-approximation for MPT?

Thank You! Questions?

Ghent, August 18, 2017 17/17