1. A Faster Algorithm for Computing the Principal Sequence of Partitions
of a Graph
Vladimir Kolmogorov
University College London

2. Graph partitioning Undirected weighted graph, weights ce  0
Cost l  0
Partition P
Minimise objective function:
[Cunningham’85]: “network attack” problem
Attacker’s goal: disconnect the network
ce : cost to break edge e
l : benefit for each additional connected component

3. Principal sequence of partitions
Increase l from 0 to +:
Nested partitions
Principal sequence of partitions [Narayanan’91]
Network strength:

4. Principal sequence of partitions
Applications:
Network reliability
Physics: computing partition function of Potts model in the infinite state limit [d’Auriac et al.’02]
Image partitioning??

5. Graph partitioning: Algorithms
Fixed l (network attack) [Narayanan’91,Baranona’92,d’Auriac et al.’02]
l1 (network strength) [Cheng&Cunnigham’94 (memory!)] , [Gabow’98]
All l‘s (principal sequence) [Narayanan’91]
n T
n T*
O(n2) T
n T*
THIS WORK
n : # of nodes
m : # of edges
T : maxflow algorithm
T* : parametric maxflow

6. Parametric maxflow Minimise for all l’s: Nested sequence of solutions
Terms Vuv(xu, xv) – submodular
Unary terms Au(l): non-decreasing functions of l
Nested sequence of solutions
[Gallo,Grigoriadis,Tarjan’89]: same worst-case complexity as for a single maxflow - O(nm log(n2/m))

7. Graph partitioning: Algorithms
Fixed l (network attack) [Narayanan’91,Baranona’92,d’Auriac et al.’02]
l1 (network strength) [Cheng&Cunnigham’94 (memory!)] , [Gabow’98]
All l‘s (principal sequence) [Narayanan’91]
Outline:
Part I: review of [Barahona’92] (fixed l)
Part II: generalising [Barahona’92] (fixed l)
Part III: parametric algorithm (all l’s)
n T
n T*
O(n2) T
n T*
THIS WORK

8. Part I: Review of [Barahona’92] algorithm
Solving network attack for fixed l

9. Relation to set minimisation
Cost of partition P:
Cost of subset A V :

10. Polymatroid P( f l) [Edmonds’70]
Set of linear lower bounds:
Non-empty subset A is tight for yP( f l) if

11. all subsets AP are tight
Optimality condition
yP( f l) and
all subsets AP are tight
 P minimises E l()

12. Greedy algorithm [Edmonds’70]: compute maximal vector yP( f l)
Init: pick yP( f l)
For each node vV:
Increase yv as much as possible v

13. Greedy algorithm [Edmonds’70]: compute maximal vector yP( f l)
Init: pick yP( f l)
For each node vV:
Increase yv as much as possible v A

14. Updating P Init: pick yP( f l), For each node vV:
Increase yv as much as possible

15. Summary Introduced: Greedy algorithm: compute maximal vector yP( f l)
Submodular function f l
Polymatroid P( f l)
Tight subsets A for yP( f l)
Greedy algorithm: compute maximal vector yP( f l)
Invariants:
yP( f l)
Subsets A  P are tight
In the end P complete  global minimum
n min cuts

16. Part II: Generalising greedy algorithm

17. Non-greedy algorithm Init: pick yP( f l) For each node vV:
[optional]
Correctness: see paper
yP( f l)
Subsets A  P are tight A

18. Part III: New parametric algorithm Solving network attack for all l‘s

19. Overview Maintain families {yl}l and {Pl}l for all l  0
Need to solve parametric problem for all l’s
[optional]

20. Naïve idea: use greedy algorithm
Does not quite work...
Functions yvl – piecewise-linear with many breakpoints
Non-monotonic!

21. [Cheng&Cunningham’94] Decreasing sequence l1 > l2 > ... > lk
Solve for li (n min cuts), use final y as initialisation for li+1
k maxflow problems for each node u are parametric
To exploit it, need to store n graphs of size O(n+m)!
Can compute network strength, but not principal sequence

22. Use non-greedy algorithm!
Functions monotonic piecewise-linear with one breakpoint
[optional]

23. Modifying v v v v u u u u

24. Modifying v v v v u u u u

25. Special case: node v v v v v

26. Correctness Need to show: for each l – non-greedy algorithm
Claim 1: updating y l preserves optimality of Al
Claim 2:

27. Claim 1 Family {Al}l is an optimal solution of
for modified family {yl}l v v v v

28. f l(.)-y l(.) : submodular
Claim 1
f l(.)-y l(.) : submodular
y lu : stay monotonic
 nestedness v v v v u u u u

29. Conclusions & Discussion
Computing principal sequence: improved complexity
Practical speed: remains to be seen
O(n2) T
n T*
Graph/image partitioning?
Potential problem: regions of different sizes
Different l‘s for different parts of the graph?
Preprocessing: initial weight => edge threshold
same or larger

31. Graph partitioning & submodularity
X: subset of edges
k(X)=# connected components in (V, X)
Supermodular
Function is submodular
Can be minimised in polynomial time
Minimum corresponds to “valid” partition P

32. Updating partitions:

33. Claim 2
for modified family {yl}l v v v v