An Intelligent Example-Generator for Graph Theory Michaela Heyer - - PowerPoint PPT Presentation

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An Intelligent Example-Generator for Graph Theory Michaela Heyer - - PowerPoint PPT Presentation

Mar 07, 2024 143 likes •902 views

An Intelligent Example-Generator for Graph Theory Michaela Heyer Boole Centre for Research in Informatics University College Cork Ireland The Problems Graph Theory relies heavily on the use of examples to help prove or disprove new

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An Intelligent Example-Generator for Graph Theory

Michaela Heyer Boole Centre for Research in Informatics University College Cork Ireland

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The Problems

Graph Theory relies heavily on the use of examples to help prove or disprove new conjectures

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The Problems

Graph Theory relies heavily on the use of examples to help prove or disprove new conjectures

  • but graphs can be quite large, so examples are

hard to find

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The Problems

Graph Theory relies heavily on the use of examples to help prove or disprove new conjectures

  • but graphs can be quite large, so examples are

hard to find Many areas outside Mathematics use graphs

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The Problems

Graph Theory relies heavily on the use of examples to help prove or disprove new conjectures

  • but graphs can be quite large, so examples are

hard to find Many areas outside Mathematics use graphs

  • but people in those areas might not have sufficient

mathematical knowledge to perform proofs of ex- istence

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The Solution

Create an easy-to-use tool which:

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The Solution

Create an easy-to-use tool which:

  • takes in a combination of graph properties
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The Solution

Create an easy-to-use tool which:

  • takes in a combination of graph properties
  • returns one or more examples of the desired graph
  • r outputs that the desired graph has not been

found

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The Approach

Two main parts:

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The Approach

Two main parts:

  • generate all graphs up to a given size
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The Approach

Two main parts:

  • generate all graphs up to a given size
  • select those with the required properties
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Generating the Graphs

The Challenge: There are lots of graphs!

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Generating the Graphs

The Challenge: There are lots of graphs! n n-node graphs 5 34 6 156 7 1,044 8 12,346 9 274,668 10 12,005,168 11 1,018,997,864 12 165,091,172,592

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Generating the Graphs

The Challenge: There are lots of graphs! But luckily...

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Generating the Graphs

The Challenge: There are lots of graphs! But luckily...

  • graphs can be generated very quickly
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Generating the Graphs

The Challenge: There are lots of graphs! But luckily...

  • graphs can be generated very quickly
  • a program for this already exists

(’nauty’ by Brendan McKay)

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Selecting the Graphs

Should be fast!

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Selecting the Graphs

Should be fast! Suppose that we are looking for a graph with properties P1, P2, . . . , Pr

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Selecting the Graphs

Should be fast! Suppose that we are looking for a graph with properties P1, P2, . . . , Pr Idea: Check P1, P2, . . . , Pr simultaneously using parallel processors

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Selecting the Graphs

Should be fast! Suppose that we are looking for a graph with properties P1, P2, . . . , Pr Idea: Check P1, P2, . . . , Pr simultaneously using parallel processors For each given graph, let ti denote the time to test for property Pi

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Selecting the Graphs

Should be fast! Suppose that we are looking for a graph with properties P1, P2, . . . , Pr Idea: Check P1, P2, . . . , Pr simultaneously using parallel processors For each given graph, let ti denote the time to test for property Pi TIME:

  • successful

test : max { ti | 1 ≤ i ≤ r }

  • unsuccessful test : min { ti | 1 ≤ i ≤ r & Pi FAILS }
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Selecting the Graphs cont’d

Observation: many graphs will fail rather than pass

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Selecting the Graphs cont’d

Observation: many graphs will fail rather than pass ⇒ the quicker we fail, the quicker we can move on

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Selecting the Graphs cont’d

Observation: many graphs will fail rather than pass ⇒ the quicker we fail, the quicker we can move on ⇒ the more properties we have, the higher the chance of a quick fail

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Selecting the Graphs cont’d

Observation: many graphs will fail rather than pass ⇒ the quicker we fail, the quicker we can move on ⇒ the more properties we have, the higher the chance of a quick fail Idea: Use an expert system to infer additional properties from the initial list

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An “Intelligent” Parallel Solution

From the required properties P1, P2, . . . , Pr infer additional derived properties Pr+1, . . . , Ps

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An “Intelligent” Parallel Solution

From the required properties P1, P2, . . . , Pr infer additional derived properties Pr+1, . . . , Ps Assume that s processors are available

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An “Intelligent” Parallel Solution

From the required properties P1, P2, . . . , Pr infer additional derived properties Pr+1, . . . , Ps Assume that s processors are available For each newly-generated graph, test properties P1, P2, . . . , Pr, Pr+1, . . . , Ps in parallel

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An “Intelligent” Parallel Solution

From the required properties P1, P2, . . . , Pr infer additional derived properties Pr+1, . . . , Ps Assume that s processors are available For each newly-generated graph, test properties P1, P2, . . . , Pr, Pr+1, . . . , Ps in parallel TIME:

  • successful

test : max { tj | Pj is the fastest test for Pi & 1 ≤ i ≤ r }

  • unsuccessful test : min { ti | 1 ≤ i ≤ s & Pi FAILS }
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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B :

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B : Add ¬ B as a fail property

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B : Add ¬ B as a fail property B ⇒ A :

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B : Add ¬ B as a fail property B ⇒ A : Add B as a pass property for A

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B : Add ¬ B as a fail property B ⇒ A : Add B as a pass property for A A ⇔ B :

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The Expert System

Apply theorems to the initial properties P1, . . . , Pr to infer additional properties Pr+1, . . . , Ps Suppose we are looking for a graph with property A A ⇒ B : Add ¬ B as a fail property B ⇒ A : Add B as a pass property for A A ⇔ B : Add B as a pass property for A and ¬ B as a fail property

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Overview of the System

List of Theorems Expert System Properties to Test Query: Required Properties Cluster: Run Property-Testing Algorithms in Parallel Output: Graph with the Required Props OR Fail Graph-Generating S/W Generated Graphs

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Example of Intelligent Solution

Query: We are looking for a graph that is critical and Hamiltonian

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Example of Intelligent Solution

Query: We are looking for a graph that is critical and Hamiltonian Required Properties: P1 : G is critical P2 : G is Hamiltonian

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Infer additional properties...

Some of the Derived Properties in Wish list:

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Infer additional properties...

Some of the Derived Properties in Wish list: P3 : α < connectivity P4 : G is a hyper cube P5 : δ ≥ v/2

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Infer additional properties...

Some of the Derived Properties in Wish list: P3 : α < connectivity P4 : G is a hyper cube P5 : δ ≥ v/2 Some of the Derived Properties in Disprove list: P6 : G is not a block P7 : G is not connected P8 : G is not biconnected P9 : G is bipartite & has bipartition(XY),|X| != |Y|

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Check Graphs...

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Check Graphs...

P1 P13 P14 P21

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Check Graphs...

P1 P13 P14 P21 is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True Not critical Discard graph! is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

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Check Graphs...

P1 P13 P14 P21

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Check Graphs...

P1 P13 P14 P21 is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True Not critical Discard graph! is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

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Check Graphs...

P1 P13 P14 P21

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Check Graphs...

P1 P13 P14 P21 is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True Discard graph! is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected Not Hamiltonian

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Check Graphs...

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Check Graphs...

P1 P13 P14 P21

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Check Graphs...

P1 P13 P14 P21 is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True...Wait... is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

P1 P13 P14 P21 True True...Wait... is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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Check Graphs...

Critical and Hamiltonian: Keep graph! P1 P13 P14 P21 True True...Wait... is critical alpha < connectivity is hyper cube is not a block is not connected is not biconnected

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The Current System

Expert system written in Haskell

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The Current System

Expert system written in Haskell Cluster code written in C++ using MPI

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The Current System

Expert system written in Haskell Cluster code written in C++ using MPI Runs on the BCRI Compute Cluster

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The Current System

Expert system written in Haskell Cluster code written in C++ using MPI Runs on the BCRI Compute Cluster Collection of approx. 200 theorems

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The Current System

Expert system written in Haskell Cluster code written in C++ using MPI Runs on the BCRI Compute Cluster Collection of approx. 200 theorems

  • Approx. 180 algorithms implemented in C++ using

LEDA

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The Current System

Expert system written in Haskell Cluster code written in C++ using MPI Runs on the BCRI Compute Cluster Collection of approx. 200 theorems

  • Approx. 180 algorithms implemented in C++ using

LEDA Some debugging still needs to be done...

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Possible Extensions

Screen for inconsistent sets of required properties

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Possible Extensions

Screen for inconsistent sets of required properties Allow for numeric comparisons: Looking for graph with e ≤ 4 Have theorem: A ⇒ e = 2 System should infer property A into wish list

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Possible Extensions

Screen for inconsistent sets of required properties Allow for numeric comparisons: Looking for graph with e ≤ 4 Have theorem: A ⇒ e = 2 System should infer property A into wish list Extend to deal with directed graphs

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Possible Extensions

Screen for inconsistent sets of required properties Allow for numeric comparisons: Looking for graph with e ≤ 4 Have theorem: A ⇒ e = 2 System should infer property A into wish list Extend to deal with directed graphs Extend to other areas of discrete mathematics