Ant Colony Optimization and the Minimum CutProblem

Timo Kotzing, Per Kristian Lehre,Frank Neumann, Pietro S. Oliveto

March 25, 2010

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Ant Colony Optimization (ACO)

I We want to analyze the use of Ant Colony Optimization(ACO) for the Minimum Cut Problem.

I As input, the ACO algorithm gets an weighted undirectedgraph G on n vertices.

I The ACO algorithm iteratively computes partitions of G ’svertices into two non-empty sets, one per iteration.

I The algorithm keeps track of the best so far candidatesolution.

I We analyze the random variable of the number of iterationsrequired until an optimal solution is found.

2/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Idea for Constructing Solutions

I Idea for Constructing Solutions (Karger and Stein):

I Any forest of n − 2 edges constitutes a partition into two sets(the sets of vertices of the two trees).

I Karger and Stein give an algorithm with expected runtimeO(n2).

I Our ACO algorithm lets ants choose (sequentially) n − 2edges to build candidate solutions (without creating cycles).

I The probability for an edge e to be picked depends on twovalue associated with that edge:

I its weight w(e); andI the pheromone value τe on e.

3/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Pheromones

I Pheromones are additional information on the edges.

I A higher pheromone value on an edge e means that e is morelikely to be chosen for the next solution.

I Initially, all pheromone values are the same.

I After that, the pheromone value of an edge e that is used inthe best-so-far solution has a pheromone value h.

I All others have a pheromone value l .

4/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Heuristic Information vs. Pheromone Values

I Remember: The probability for an edge e to be pickeddepends on the two values w(e) and τe .

I How do we balance these two values?

I We use two parameters, α and β.

I For an edge e with associated pheromone value τe and weightw(e), the ant chooses e proportionally to

ταe · (w(e))β.

5/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Results

ταe · (w(e))β

We proved the following expected optimization times of ACO.

I If α = 0 and β = 1 (greedy only), O(n2).

I If α = 1 and β = 1 with constant pheromone bounds, timesare still polynomially bounded.

I If α = 1 and β = 1 with at least linear pheromone boundratio, times are not polynomially bounded.

I If β > 1, times are not polynomially bounded.

I If α = 1 and β = 0 for sensible pheromone bounds, times aregain not polynomially bounded.

6/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Conclusions

I Don’t use an ACO algorithm to solve the Min-Cut Problem.

I ACO can simulate Karger and Stein’s algorithm.

I We now understand better how ACO algorithms work.

I We now understand better how to analyze ACO algorithms.

7/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Conclusions

I Don’t use an ACO algorithm to solve the Min-Cut Problem.

I ACO can simulate Karger and Stein’s algorithm.

I We now understand better how ACO algorithms work.

I We now understand better how to analyze ACO algorithms.

7/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Conclusions

I Don’t use an ACO algorithm to solve the Min-Cut Problem.

I ACO can simulate Karger and Stein’s algorithm.

I We now understand better how ACO algorithms work.

I We now understand better how to analyze ACO algorithms.

7/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Conclusions

I Don’t use an ACO algorithm to solve the Min-Cut Problem.

I ACO can simulate Karger and Stein’s algorithm.

I We now understand better how ACO algorithms work.

I We now understand better how to analyze ACO algorithms.

7/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Conclusions

I Don’t use an ACO algorithm to solve the Min-Cut Problem.

I ACO can simulate Karger and Stein’s algorithm.

I We now understand better how ACO algorithms work.

I We now understand better how to analyze ACO algorithms.

7/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut

Thank you.

8/8 Kotzing, Lehre, Neumann, Oliveto ACO and MinCut