When Ants Attack! Ant Algorithms for Subset Selection Problems

1. When Ants Attack!Ant Algorithms for Subset Selection Problems

2. Overview • Ant Algorithms for Best Hamiltonian Path-Finding Problems • Subset Selection Problems • Definitions and examples • Ant-SS, an Ant Algorithm for Subset Selection Problems • Solution construction step • Pheromone updating step • Parameters • Experimental Results • Conclusions

3. Ant Algorithms • Find solutions or near-solutions to intractable discrete optimization problems • Inspired by the behaviour of real ants • Ants communicate indirectly by laying trails of pheromone • “stigmergy” • Enables the ants to find shortest paths between nest and food

4. Ant Algorithms • Problem is modelled as a search for a minimum cost path in a graph • Artificial ants complete a series of walks of the graph • Each path corresponds to a potential solution • They deposit pheromone on their path in proportion to its quality • Unlike real ants, typically • the pheromone is deposited on completion of the path • only the best paths receive pheromone (“elitism”)

5. Ant Algorithms • The ants choose between destinations probabilistically based on a combination of • the cumulated pheromone τ, and • an heuristic factor η η<A,B> Τ<A,B> B A Τ<A, C> η<A,C> C

6. Ant Algorithms • The pheromone intensifies search around the most promising paths • But to avoid stagnation, we need techniques to diversify search, e.g: • raise the importance of the heuristic factor • evaporate some pheromone • impose lower and upper bounds on the pheromone (MAX-MIN Ant Systems)

7. Travelling salesperson problem • Find the minimum length route that: • starts in, e.g., Cork; • visits each town exactly once; • returns to Cork.

8. And now the bad news... 7 360 360 microsecs 3 billion (10-digit number) 14 52 mins Nearly 40,000 years A 19-digit number 21 2 trillion centuries 28 A 28-digit number

9. Demo

10. Best Hamiltonian Path-Finding Problems • Ant algorithms have been used for • Travelling salesperson problems • Quadratic assignment problems • Vehicle routing problems • Permutation constraint satisfaction problems • Characteristics • ants must visit every vertex in the graph • heuristic factor is often inverse of edge cost

11. Subset Selection Problems • Intuition: • Given a set S, find a subset S’  S • that satisfies certain properties and/or • optimizes some objective function • Example: the multidimensional 0-1 knapsack problem • Choose which objects to put into the knapsack • but total weight < 15kg (and perhaps other properties) • and maximise monetary value

12. Subset Selection Problems • An SS problem may be defined by a triple (S, Sfeasible, f) where • S is a set of objects; • Sfeasible (S) is a set that contains all feasible subsets of S; • f : Sfeasible   is an objective function that associates a real-valued cost f(S’) with every feasible subset of objects S’  Sfeasible. • The goal of an SS problem (S, Sfeasible, f) is to find S*  S such that S*  Sfeasible and f(S*) is maximal.

13. Multidimensional knapsack as a SS Problem • S is the set of objects; • Sfeasible contains every subset that satisfies all the resource constraints, i.e. Sfeasible = {S’  S | i1..m, jS’, rij < bi} where m is the number of resources, rij is the consumption of resource i by object j, and bi is the available quantity of resource i; • f returns the total profit, i.e., S’Sfeasible, f(S’) = jS’ pj where pj is the profit associated with object j.

14. Other SS Problems • Maximum Clique • Maximum Boolean Satisfiability • Maximum Constraint Satisfaction • Minimum Vertex Cover • Graph Matching • Edge-Weighted k-Cardinality Tree

15. Ant-SS • Initialise pheromone • repeat • for each ant k, construct a solution Sk as follows: • Randomly choose a first object oi S • Sk  {oi} • Candidates  {oj  S \ oi | Sk  {oj}  Sfeasible} • whileCandidates   • Choose an object oi  Candidates with probability p(oi, Sk) based on τ(oi, Sk) and η(oi, Sk) • Sk  Sk  {oi} • Remove oi from Candidates • Remove from Candidates every object oj such that Sk  {oj}  Sfeasible • end while • end for • Optionally, apply local search • Update the pheromone • until maximum number of cycles reached or acceptable solution found Sfeasibe must be defined in such a way that feasible subsets can be constructed incrementally

16. Pheromone update • In SS Problems, the order in which objects are selected is not significant • hence rewarding consecutively visited vertices (by placing pheromone on visited edges) is not meaningful • Vertex pheromone strategy • reward solution Sk by associating pheromone with each object oi Sk • Clique pheromone strategy • reward solution Sk by associating pheromone with each pair of different objects (oi, oj)  Sk  Sk

17. Vertex Pheromone Strategy • Pheromone is laid on objects • represents the learned desirability of selecting that object The ant’s walk Where pheromone is laid 3 3 2 2 1 4 1 4 5 5

18. How does this affect candidate selection? • Initialise pheromone • repeat • for each ant k, construct a solution Sk as follows: • Randomly choose a first object oi S • Sk  {oi} • Candidates  {oj  S | Sk  {oj}  Sfeasible} • whileCandidates   • Choose an object oi  Candidates with probability p(oi, Sk) based on τ(oi, Sk) and η(oi, Sk) • Sk  Sk  {oi} • Remove oi from Candidates • Remove from Candidates every object oj such that Sk  {oj}  Sfeasible • end while • end for • Optionally, apply local search • Update the pheromone • until maximum number of cycles reached or acceptable solution found

19. Vertex Pheromone Strategy: how it affects candidate selection Suppose the ant has selected objects 1, 2 and 4. • The probability of selecting • object 5 is based on • the pheromone on object 5; • the heuristic value of object 5 3 3 2 2 1 4 1 4 5 5 Τ5 η5

20. Clique Pheromone Strategy • Pheromone is laid on pairs of objects • represents the learned desirability of selecting one object given that another has been selected The ant’s walk Where pheromone is laid 3 3 2 2 1 4 1 4 5 5

21. Clique Pheromone Strategy: how it affects candidate selection Suppose the ant has selected objects 1, 2 and 4. • The probability of selecting • object 5 is based on • the pheromone between objects • 1 & 5, 2 & 5, and 4 & 5; • the heuristic value of object 5 3 3 2 2 1 4 1 4 5 5 Τ5 = Τ(1,5) + Τ(2, 5) + Τ(4, 5) η5

22. Comparison • Vertex Pheromone Strategy • evaporation: O( |S| ) • reward of subset Sk: O( |Sk| ) • Clique Pheromone Strategy • evaporation: O( |S|2 ) • reward of subset Sk: O( |Sk|2 )

23. Experiments • τmin = 0.01; τmax = problem-dependent • Evaporation rate, ρ = 0.01 • Number of ants = 30 • Pheromone factor weight = 1 • Heuristic factor = problem-dependent • Heuristic factor weight = problem-dependent • Pheromone update strategy: tried both • Local search: tried without/with

24. Maximum Clique Problem • τmax = 6 • Heuristic factor, η = 1 (in effect, none) • Heuristic factor weight = 0 • Results averaged over 50 runs

25. Solution Quality Results • All variants find optimal solutions to ‘easy’ instances • On harder instances, Clique outperforms Vertex • Incorporating local search improves solution quality

26. Solution Time Results • Vertex needs fewer cycles than Clique • Hence, and because it is cheaper, Vertex converges in less time than Clique • Local search reduces Cycles but not always Time

27. Convergence (C500.9)

28. Comparisons with other algorithms • In other experiments on maximum clique problems (Solnon), Ant-SS • outperforms a genetic algorithm • is comparable to an adaptive greedy search • is slightly outperformed by a tabu search

29. Constraint Satisfaction Problems • τmax = 4 • Heuristic factor η is • select variable based on smallest-domain • heuristic factor of variable’s value is inversely proportional to the number of additional constraints that would be violated • Heuristic factor weight = 10 • Results averaged over 100 runs

30. Solution Quality Results • All variants find solutions to ‘easy’ instances • On harder instances, Clique outperforms Vertex • Incorporating local search improves solution quality • All instances are solvable. • We show % of runs that succeeded in finding a solution

31. Solution Time Results • Clique needs fewer cycles than Vertex!!! • But, because it is cheaper, Vertex converges in less time than Clique • Local search reduces Cycles but not always Time

32. Comparisons with other algorithms • In other experiments on constraint satisfaction problems (Solnon), Ant-SS • outperforms a genetic algorithm because Clique nearly always finds a solution even on harder instances, but the genetic algorithm does not • is slower than a complete search on smaller instances but much faster on larger instances

33. Conclusions • We defined Subset Selection Problems • We have defined the Ant-SS algorithm • with two pheromone strategies, Vertex and Clique • We have evaluated on a number of problem instances • good performance relative to other algorithms • Clique typically finds better quality solutions but Vertex generally requires less CPU time