The ACO Metaheuristic

1. The ACO Metaheuristic ACO 2.1 - 2.3 January 2008 C. Colson

2. Ants in Mythology • Zeus turned the hardworking ants of the uninhabited island of Aegina into the subjects of Aeacus. The people were called the Myrmidons. • Pēleús (son of Aeacus) was the father of Achilles • Although Achilles was from Thessaly, he leads his distant kinsmen, the Myrmidons in the Trojan War as told by Homer in the Iliad. • Of the Myrmidons: “Industry, thrift, endurance; they are eager for gain, and never easily relinquish, what they have won!"

3. Metaheuristic? • Refers to a master strategy that modifies sub-heuristics. • A general purpose heuristic designed to guide an underlying problem-specific heuristic towards promising solutions. • NP-hard problems are believed to be unsolvable in polynomial time (tractable and intractable problems). • As a result, heuristic methods are applied to get near-optimal results in a reasonable time. • Metaheuristics have entered the picture to guide “heuristic methods” applicable to widely varying problems. •    ACO is a metaheuristic framework: applicable to many different applications.

4. Combinational Optimization • Involve finding values for discrete variables such that an optimal solution with respect to a given objective function is found. • Traveling salesman • Supply-chain logistics planning • Asset allocation • Maximize or Minimize applications: П(S, f, Ω) • Problem: general question to be answered • Instance: a case of specified values for a problem • S: set of candidate solutions • f(s): objective function (s Є S) • Ω: constraints • Feasible solutions: Ŝ subset of S, that satisfies Ω • Globally optimal solution: s* Є Ŝ • Minimization example: f(s*) ≤ f(s) for all {s Є Ŝ} • Maximization example: f(s*) ≥ f(s) for all {s Є Ŝ}

5. Computational Complexity • Straight-forward approach to solution: exhaustive search! • The possible solutions grows exponentially with instance size (n). • Time complexity: maximum time an algorithm needs to find a solution to an instance of n-size (aka worst-time complexity). • “Big-O” formal notation: Ŏ(function) • A function g(n) is said to be Ŏ( h(n) ), if two positive constants A and n0 exist such that g(n) ≤ Ah(n) for all n≥n0 . This is the asymptotic upper bound. • Polynomial time complex: if Ŏ( g(n) ) where g(n) is a polynomial. If k is the largest exponent of the polynomial g(n), then the problem is said to be solvable in Ŏ( nk ) time. • Exponential time complex: if Ŏ( g(n) ) cannot be bounded by a polynomial. • Intractable = not polynomial time complex.

6. NP-hard??? NP-Completeness (Bottom of page 28) • P-class: an algorithm that outputs the correct answer (“yes” or “no) in poly-time. • NP-class (stands for Non-deterministic Polynomial): an algorithm that verifies every instance is indeed “yes” in poly-time. • P is a subset of NP • Poly-time reductions: the transformation of one problem into another one by a poly-time algorithm. • Key point: if the resultant problem is solvable in poly-time, then the original problem is likewise solvable in poly-time! NP Problems P = NP-complete ??? P Problems NP-complete Problems

7. Two more classifications of algorithms: • Exact : • guaranteed to find optimal solution • prove optimality for every (finite-size) instance • runs within instance-dependent time (worst case scenario: exponential-time) • Approximate (trades optimality for speed/efficiency) heuristic methods; seeks near-optimal solutions but cannot guarantee optimality. • Further classified into constructive and local searches • Constructive (iteratively add solution components to the empty solution set until solution set is found): incremental solutions without backtracking (see NNH for TSP problem on pg. 30) • Local search (starts from a full solution set and tries to make improvements by local changes): iterative exploration that seeks to improve the solution with local changes (see best-improvement rule for neighborhood examination scheme on pg. 31) • Neighborhood structure: the set of possible solutions that the algorithm can “move” to from the current solution.

8. The ACO Metaheuristic • A colony of artificial ants cooperate to find good solutions to difficult discrete optimization problems. • Good solutions are an emergent property of cooperation! • Static problems: all characteristics of the problem are defined once and do not change. • Dynamic problems: characteristics vary according to underlying functions and the optimization must adapt to the changing environment. • See problem definition on pgs. 34 & 35.

9. ACO (continued)… • Ants build solutions by performing stochasic action on the construction graph which is made up of components (nodes) and connections (paths). • Sometimes the ants find feasible solutions, sometimes not (and that’s ok). • The pheromone trail is coded long-term memory. • Heuristic information: additional information that the ants have, a priori, from a source other than the environment (example: estimated path cost) • Although ants act concurrently, independently, and most times dumbly, good–quality solutions arise from collective interaction of the ants.

10. ACO Components • ConstructAntsSolutions: • manages ants concurrently • stochastic engine resides here • evaluation function for ant performance resides here • UpdatePheromones: • deposits or evaporates pheromone trails. • DaemonActions: • centralized actions not resident in individual ants • optional functionality • example: pheromone bonuses for shortest path yet • See page 38 for pseudocode.

11. ACO Applications • Hamiltonian circuit: a trip solution on an “undirected” graph which visits each city (node) exactly once and returns to the starting city. • Symmetric TSP: • only paths have “cost” • shortest Hamiltonian circuit trip length • “cost” from node i to j is identical to j to i. • Asymmetric traveling salesman problem • “cost” from node i to j is not identical to j to i. • Sequential Ordering Problem: similar to TSP • an asymmetric traveling salesman problem with additional constraints • doesn’t need to return to starting city • a precedence constraint must be considered • the precedence constraint requires that some node i has to be visited before some other node j • Note: pheromones play roughly the same role in these cases.

12. ACO Applications (continued…) • Generalized Assignment Problem: • a number of agents and a number of tasks • any agent can be assigned to perform any task, but incurs some cost/profit that varies with the assignment • each agent has a budget and the sum of the costs of task assigned (cannot exceed its budget). • solution is an assignment in which all agents do not exceed their budget • good solution minimizes cost or maximizes “profit” • pheromones associated with either the next task to choose or which agent to assign the task to. • Multiple Knapsack Problem: • goal is to maximize valuable items that can fit into one bag to be carried on a trip • given a set of items, each with a cost and a value, determine the number of each item to include in a collection so that the total cost is less than a given limit and the total value is as large as possible • pheromones are associated only with the desirability of adding a particular item to a solution

13. ACO Applications (continued…) • Network Routing Problem: • minimum path costs between pairs of nodes in the network • each “connection” (path) should have multiple pheromone trails associated for each different node destination. • Dynamic TSP: • time is a factor • cities can be added or removed from the graph • pheromones act similarly to standard TSP problem.