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
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Efficient Computation

• How can we make this computation fast?

12
Caching Previously Seen Graphs

• Caching previously seen graphs

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Biconnectivity

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

BC2
BC1
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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)

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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
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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)
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Future Work

• Decremental Graph Algorithms
• Decremental Graph Isomorphism ?
• Decremental Biconnected Components
• Edge Selection Heuristics
• Can we understand why Vorder and Minsdeg do well?
• Can we find better heuristics?
• Edge Addition/Contract
• Can we implement this for Tutte?
• Can we move towards other graph classes --- e.g.
  chordal graphs?

22
Edge Contract/Addition

• Contract/Addition
• T(G) T(G/e) T(G-e)
• ? T(G-e) T(G) T(G/e)
• ? T(G) T(Ge) T(G/e)
• Idea
• For dense graphs move towards complete graph
• For sparse graphs, move towards empty graph (as
  before)
• Problem
• Computing Tutte polynomial for complete
  multi-graph is difficult ?
• BUT, for chromatic polynomial computation we have
  not multigraphs