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EdgeSelection Heuristics for Computing Tutte Polynomials

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Victoria University of Wellington. Gary Haggard. Bucknell, USA. Gordon Royle. University of Western Australia. COMP205 Software Design and Engineering ... – PowerPoint PPT presentation

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Title: EdgeSelection Heuristics for Computing Tutte Polynomials

1
Edge-Selection Heuristics for Computing Tutte
Polynomials

David J. Pearce
Victoria University of Wellington
Gary Haggard
Bucknell, USA
Gordon Royle
University of Western Australia

2
Tutte Polynomial what is it?

Its a 2-variable polynomial on graphs
T(x,y) y y2 x 2xy 2x2 x3
What can we do with it?
T(1,1) gives the number of spanning trees
T(2,2) gives 2e
T(1-x,0) gives the Chromatic polynomial P(x)
T(0,1-x) gives the Flow polynomial F(x)
T(,) gives who knows?

3
Great, but why do we care?

Many applications of Tutte polynomial
Physics (q-state Potts model), Biology and
probably lots more
Knots
Tangled cords which cant be unravelled
Problem how do we know when two knots are same?

4
Computing Tutte Polynomials

Delete/Contract Operations
Tutte Definition
T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

G
Ge
G/e
5
An Example

T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

6
An Example

T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

x
7
An Example

T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

x
x
8
An Example

T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

x
x
x
x
9
An Example

T(G) 1, if G ?
T(G) xT(G-e), if e is a bridge
T(G) yT(G-e), if e is a loop
T(G) T(G-e) T(G/e), otherwise

x
x
x
x
x
x
y
x2
10
Larger (but still tiny) examples
11
Efficient Computation

How can we make this computation fast?

12
Caching Previously Seen Graphs

Caching previously seen graphs

13
Biconnectivity

Biconnected Components
If G not biconnected
Then, there is a bridge
So, maintain biconnected components of G!!
Triconnected Components

BC2
BC1
14
Edge-Selection Heuristics

Vertex Order (VORDER)
A fixed order of vertices is used
Edges from 1st node first, then 2nd, then 3rd,
etc
Minimise Single Degree (MINSDEG)
Edge selected whose end-point has smallest degree
Minimise Degree (MINDEG)
Edge selection whose degree sum is smallest of
any
Maximise Single Degree (MAXSDEG)
Maximse Degree (MAXDEG)

15
Edge-Selection Heuristics
1
2

Vertex Order (VORDER)
A fixed order of vertices is used
Edges from 1st node first, then 2nd, then 3rd,
etc
Minimise Single Degree (MINSDEG)
Edge selected whose end-point has smallest degree
Minimise Degree (MINDEG)
Edge selection whose degree sum is smallest of
any
Maximise Single Degree (MAXSDEG)
Maximse Degree (MAXDEG)

5
3
4
16
Edge-Selection Heuristics
1
2

Vertex Order (VORDER)
A fixed order of vertices is used
Edges from 1st node first, then 2nd, then 3rd,
etc
Minimise Single Degree (MINSDEG)
Edge selected whose end-point has smallest degree
Minimise Degree (MINDEG)
Edge selection whose degree sum is smallest of
any
Maximise Single Degree (MAXSDEG)
Maximse Degree (MAXDEG)

5
3
4
17
Experimental Results
18
Experimental Results
19
Random Graph (12 vertices, 20 edges)
Minsdeg (188 graphs, 47/91 hits)
VOrder (272 Graphs, 72/170 hits)
20
Random Graph (9 vertices, 16 edges)
Minsdeg (150 graphs, 14/50 hits)
VOrder (138 Graphs, 24/86 hits)
21
Future Work