1. ACO for NP-hard Problems (continued) ACO 5.4 - 5.5 February 2008 C. Colson

2. Its not easy being an ant…

3. Review from Monday’s ACO Applications to NP-hard (5.1-5.3)  Intractable: problem that is “so hard” that it cannot be solved in poly-time.  Combinational optimization problems (three primary classes): –Routing problems –Assignment problems –Scheduling problems  Each are related, but nuances of construction modify their ACO implementation  ACO has shown “world-class” performance on some combinational optimization problems, especially compared to the best available exact algorithms.

4. What we are going to consider today:  The application of ACO to: – Subset problems and –Other NP-hard problems (not already discussed)

5. What I hope to demonstrate:  That the ACO algorithm is adaptable to problems that are loosely related in structure.  How to adapt what we learned about ACO implementation in sections 5.1- 5.3 for other NP-hard problems.

6. Broadly, what is a Subset Problem?  A solution is represented by a subset of available components (subject to constraints).  Clearly, this is confusing because: –Numerous problems we have already considered could be subset problems (i.e. TSP)  That said, here are some common problems that are approached from a subset problem point-of-view:  Set covering  Weight Constrained Graph Tree Partitioning  Arc-Weighted l-Cardinality Trees  Multiple Knapsack  Maximum Independent Sets  Maximum Clique  Redundancy Allocation

7. Some comments about the Subset Problems  The Subset problem (SP) is fundamentally different than the (sequential) ordering problems. –There is no “path” concept here. –ACO cannot be applied in its direct form.  Ordering problems have fixed length, SPs do not. –The ACO implementation for SPs must establish an N max used to determine the end of the cycle for all ants.  Given a set U of n items, the subset problem selects the best subset, s, which satisfies constraints.  Ants do not leave pheromones on each edge, but instead on each element of U. This means that an element with a higher pheromone level is more profitable than others.

8. Consider this example:  Set covering: matrix of 1’s and 0’s, each column has a weight (cost). –Pheromones are stacked on the columns chosen, not the arcs “between” columns –Equation 5.12 is modified from the Eqn. 3.2 form to exhibit the objective function directly (not {tour length} -1 as with standard ACO) –One ant at a time

9. Similarities in other Subset Problems  Set covering  Weight Constrained Graph Tree Partitioning  Arc-Weighted l-Cardinality Trees  Multiple Knapsack  Maximum Independent Sets  Maximum Clique  Redundancy Allocation  Arcs & nodes have costs & weights, but the solution path must fall within a certain cost/weight range (not a complete tour)  Pheromones are attached to elements, not arcs  Pheromones attached to arcs (similar to traditional ACO)

10. Now, some “other” NP-hard problems  Shortest Common Supersequence  Bin packing  2D-HP Protein Folding  Constraint Satisfaction  Comments: –Its “lookahead” feature and the “vision” an ant has to assess multiple paths smells like GA or local search? –Ant Colony is split up into islands (i.e. share solutions only occasionally). This facilitates working the problem from both ends (empty-to-solution and solution- to-empty) –Seems very similar to Maximum Independent Sets or Maximum Clique –“Lookahead” is also used –At first glance, does not seem like a good fit for ACO –Similar to Assignment Problems –Employs a form of the Min-Conflicts heuristic

11. Bin Packing Example  Packing # of items in bins of a fixed capacity (BPP) or  Cutting items from stocks of a fixed length (CSP)  Some questions to ask yourself: –1. How can good “packings” be reinforced via the pheromone matrix? –2. How can the solutions be constructed stochastically, with influence from the pheromone matrix and a simple heuristic? –3. How should the pheromone matrix be updated after each iteration? –4. What fitness function should be used to recognize good solutions?  From the ordering problem perspective: many permutations are possible.  From the grouping problem perspective: τ(i,j) expresses the favorability of having items of size i and j in the same bin.  Pheromone matrix works on item sizes, not the items themselves.