CS 371 - Introduction to Artificial Intelligence
Homework #3

Due: Thursday 9/23 at the beginning of class
Note: Work is to be done in pairs

Stochastic Local Optimization

HW3 Code Resources

1. The Channel Routing Problem (CRP):

For this exercise, you'll be given an Annealable class by C.S. major and salutatorian David Hettlinger '04 from his research of summer 2002. This class implements a complex combinatorial optimization problem concerning VLSI circuit design. For this "Channel Routing Problem", horizontal and vertical wires are on opposite side of and connected through a silicon wafer. We seek to find a wiring layout between pins with corresponding numbers that minimizes the number of necessary horizontal tracks. All implementation of the Annealable interface is supplied for you. Your task is to implement and collect data on optimization performance with 10000 iterations for:

iterative improvement with 0%, 10%, 20%, 40%, and 100% uphill step acceptance, and
simulated annealing with an initial temperature of 70 and a temperature decay of .95 per iteration.

Construct each initial CRP state with "new CRState4("dde.txt")". All other interaction with this class will be according to the Annealable interface. Record the final energy (i.e. channel width) for each optimization run, and perform a minimum of 3 runs for each method/setting above (i.e. minimum 18 runs total). We will share results for this problem on Tuesday 9/21 in class.

2. Choose Your Own Optimization Adventures: Implement 2 of the 3 following problems using the Annealable interface and write a test program that demonstrates the effectiveness of one of our stochastic local optimization techniques for the problem. (If there are other optimization problems you would like to try, please see me.) Print the energy of the minimum energy state at intervals during and at the end of the optimization. Limit the time for each run to 1 minute. You'll want to use even shorter runs as you develop, test, and debug.

Scheduling Problem: Apply simulated annealing to the following problem: Given a number of students wishing to take a number of classes which can be assigned to a number of time slots, assign classes to time slots in such a way as to minimize schedule conflicts. You will create a class SchedulingProblem which implements Annealable. Its constructor will take a number of students, the number of classes a student takes, a number of classes offered, and a number of time slots (e.g. 240 students with 5 classes each, 100 classes offered, 12 time slots). Each randomly generated instance of the problem will associate each student with the given number of different random classes. Each class will be associated with a random time slot. For the purposes of the Annealable interface, next causes a class to be randomly reassigned to a different time slot. Energy is the sum of the student conflicts. The number of conflicts for a student is the number of their classes they cannot attend given the current schedule. (For the example above, if all of a student's classes are scheduled at the same time, the student has 4 conflicts.) With parameters that generate problems which will likely not have a zero-conflict schedule, demonstrate the performance of simulated annealing versus iterative improvement with only downhill steps.
Traveling Salesman Problem with River: (From "Numerical Recipes in C" by Press et al) Uniformly and randomly distribute n points (i.e. "cities") within a unit square. Represent a circuit (or "tour") of these cities as a permutation in an appropriate list data structure. (Assume that one returns to the first city after visiting the last city.) Compute the "energy" of a tour state as the total distance of the circuit plus a fixed cost for each time the tour crosses a north-south "river" which divides the square vertically in half. The initial state constructor should take n as a parameter. Graphically display the solution, with tour lines in black and the river line in blue. Hint: Generate each next state by taking a random segment of the tour and reversing the order (e.g. ABCDEF -> ADCBEF or ABCDEF -> FECDBA). In doing so, one need not recompute the cost of the whole tour, but only the paths that have changed.
Tree Planting Problem: Suppose one has n trees and an plot that is circular in which the trees should be planted. Further suppose that it is desired that the trees should have maximal separation from each other and from the borders of the plot. Construct the initial state with n trees positioned randomly. Compute the next state by taking a single tree and moving it by a small amount randomly (e.g. according to a small Gaussian distribution; see java.util.Random.nextGaussian()). If a tree moves outside of the plot, move it to the closest point within the plot. Experiment with different energy function choices. For each tree, compute the minimum distance to the plot borders and all other trees. Then have the energy function return some function of these distances (e.g. the negated minimum, the mean, etc.). Which energy function gives you the most satisfactory result? Graphically display the solution, showing the plot outline, and the trees as filled green circles with radii being the minimum of all the minimum distances.

Hopefully, in looking at these problems, you'll see the wide applicability of such optimization algorithms. Often, the most challenging aspect of such optimization is devising a good, efficient next state generator that gives beneficial structure to the state space, rather than merely bouncing randomly from one state to another. Good next state generators both allow one to traverse the entire state space, and immediately sample states that are similar in quality to the current state.

Todd Neller