A Connectivity Constraint using Bridges Patrick Prosser

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A Connectivity Constraint using Bridges
Patrick Prosser Chris Unsworth
An adjacency matrix model of a graph Made up of
constrained integer variables Aij

• Applications
• network design problems
• phylogenetics (bioinformatics)
• many graph theoretic problems

Allow a search process to select/reject edges
whilst ensuring the graph is connected
Yes. This is constraint programming!
NOTE If reject (4,6) force (5,6) If reject (5,6)
force (4,6) Must take (3,4) a bridge!!! (1,3)
is irrelevant
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The trick maintain a DFS tree with back-edges!
A specialised/n-ary constraint with its own data
structures and methods

• Build a DFS tree with tree edges T and back edges
  B
• Maintain counters on tree edges, the number of
  paths that use this edge
• If a counter on a tree edge falls to 0 it is a
  bridge and must be selected/forced

1
1
2
1
3
Counters on paths through tree edges
bridge
0
4
1
5
1
6
3
Actions
Select edge
Internals of constraint
The Graph
The 0/1 variables
Select edge (V,W)
4
Actions
Reject back edge
Internals of constraint
The Graph
The 0/1 variables
Reject edge (X,T)
NOTE tree edge counters decremented
5
Actions
Reject tree edge
Internals of constraint
The Graph
The 0/1 variables
Reject edge (U,V)
6
Actions
Reject tree edge (U,V)
1. Decrement tree edge counters assuming we
remove back edges From the sub tree rooted on V
3. Build new dfs tree rooted on w (T,U) has
count of 0 and is a bridge!
2. Find deepest back edge to become a new
tree edge and remove old edges in sub tree