ACO for NPhard Problems continued ACO 5 4

ACO for NP-hard Problems (continued) ACO 5. 4 - 5. 5 February 2008 C. Colson

Its not easy being an ant…

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.

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

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. 15. 3 for other NP-hard problems.

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

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 Nmax 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.

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

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)

Now, some “other” NP-hard problems § § Shortest Common Supersequence Bin packing 2 D-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 solutionto-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

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.