Optimization Problems

Optimization Problems Greg Stitt ECE Department University of Florida

Introduction • Why are we studying optimization problems? • Just about every RC-tool related issue is an optimization problem • This lecture should provide abstract solutions to many RC tool problems

Time Complexity • Time Complexity - Informal definition: • Defines how execution time increases as input size increases • We are interested in worst case execution time • Big O Notation • O(n) • Execution time increases linearly with input size • O(n2) • Quadratic growth • O(2n) • Exponential growth

Problem Classes • P • A problem is “in P” if it has a polynomial time solution • O(n), O(n2), O(n3), etc. • Undecidable • Provably impossible to solve • Example: halting problem • NP • Does not mean “Not-Polynomial”!!!! • A NP problem has a non-deterministic polynomial time solution • NP-hard • A problem is NP-hard if every problem in NP is reducible to it • Basically means that NP-hard problems are at least as hard as the hardest problems in NP • NP-complete • NP + NP-hard • Most interesting problems are NP-complete!!!! • Traveling salesman, Subset sum, 0-1 knapsack, graph coloring, vertex cover • Place and route, logic minimization, minimum resource scheduling, hw/sw partitioning, etc.

The Big Question • Does P = NP? • Been studied for a long time • Never been proven either true or false • Why do we care? • Currently, best known solutions for NP-complete problems typically have complexity of O(2n), O(n!) • Known as intractable – takes more than your lifetime to solve • If one NP-complete problem can be solved in polynomial time (is in P), then all NP-complete problems can be solved in polynomial time • Remember, many interesting problems are NP-complete • If you can find a polynomial time solution to an NP-complete RC problem: • I’ll give you an A And, I’ll get a million dollars (www.claymath.org/millennium/P_vs_NP/ )

Problem Classes

Optimization Problems • Informal definition • Problem of finding the best solution from all possible solutions • Typically involves finding best solution for millions, billions, or even more possibilities • Possible solution: exhaustive search • Check every possible solution, save best one • Works, but • Many optimization problems are NP-complete • Not feasible to generate all possible solutions • Example: O(2n) • n = 5 => 32, n = 10 => 1024, n = 100 => 1.3 * 1030 • Problem: If most optimization problems are intractable, how do we solve them? • Could use solution to any NP-complete problem! • Remember: any NP-complete problem can be reduced to any other NP-complete problem • But, this doesn’t really help • There is no known polynomial solution to any NP-complete problem • So, what can we do?

Branch and Bound • Another solution: • Branch and bound • Idea: Try to eliminate many possibilities without evaluating them • How it works: • Imagine a tree representing all possible solutions • Algorithm progressively builds tree • Maintains best found solution so far • When considering a branch in the tree, determines best possible solution represented by branch • If branch can’t possibly be better than current best, don’t both to explore possible solutions • “Prunes” solution space • Result • + Very often, can eliminate large percentage of possible solution • + Still finds optimal solution • + Usually, much faster than exhaustive search • - But, still has exponential complexity  • O(2n) • - Still not feasible for large input sizes

Heuristics • Another solution: • Forget about trying to find the best solution • Focus on finding a “good” solution quickly • Known as heuristics • Good candidates for heuristics • Map problem to other NP-complete problem, use heuristic for that problem • Source of way too many publications • Use generalized optimization problem heuristic • Due to large number of interesting optimization problems, research has introduced general heuristics that apply to all optimization problems • Examples: • Hill Climbing • Simulated Annealing • Genetic Algorithms

Hill Climbing • Background: Graph of solution space • x axis = possible solutions • y axis = quality of solution • Height of solution represents goodness • Peak represents best solution • Informal description: • 1) Choose a solution s (usually randomly) • 2) Find neighboring solution n (i.e. change 1 thing) • 3) If n better than s (climbs the hill) • s = n, Repeat from 2) • 4) If all neighboring solutions worse than s • s is answer (s is a peak) • Example: Traveling Salesman • 1) Find a solution • 2) Swap 2 cities, if improves solution keep, if not reject • 3) repeat 2) until no improvement can be found

Hill Climbing • Advantages: • Very fast, works well for certain problems • Disadvantages: • What if there are multiple peaks? • Hill climbing gets stuck at all peaks, known as local maxima • Optimal solution is highest peak – global maximum • May result in extremely suboptimal solution if many peaks

Simulated Annealing • Problem: Hill climbing suffered from local peaks • Need way of escaping local peaks • => Have to accept some bad moves • Solution: Simulated annealing • Annealing is process of heating and cooling metals in order to improve strength • Idea: Controlled heating and cooling of metal • When hot, atoms move around • When cooled, atoms find configuration with lower internal energy • i.e. makes metal stronger • Analogy: • Temperature = probability of accepting worse neighboring solution • When temperature is high, likely to accept worse neighboring solutions (but may lead to better overall solution) • Analogous to atoms wandering around • Cooling represents shrinking probability of accepting worse solutions

Simulated Annealing • Informal description • 1) Set initial temperature and probability p • 2) Find initial solution s • 3) Find neighboring solution n • 4) If n better than s • s = n (always accept better solutions) • 5) If n worse than s • Accept n with probability p • 6) Reduce “temperature” and p by some amount • 7) If final temperature not reached, repeat from 3) • 8) If final temperature reached, report best solution found

Simulated Annealing • Advantages • Can find near optimal solutions – escapes local maxima • Disadvantages • Takes a long time to find near optimal solution • But, still fast compared to algorithms • Very sensitive to input parameters – must be configured well to get good results • How long to run? • Initial temperature/final temperature • What should initial probability be? • Cooling schedule • How much to reduce temperature and probability at each step?

Genetic Algorithms • Another possible solution: • Imitate evolution – survival of the fittest • Assumption: evolution produces “better” humans/animals • Implies genetic processes must work well • Genetic Algorithms • Generates a “population” of solutions • Selection chooses members of population (solutions) to survive by probability based on fitness function (i.e. natural selection) • Bad solutions less likely to “survive” • Reproduction combines attributes of selected population • Crossover/Inheritence – Combines traits for each parent (solution) • Mutation – Random change to characteristic • Repeat until solution has “evolved” to acceptable lavel

Genetic Algorithms • Advantages • Can find near optimal solutions • Disadvantages • Takes a long time to find near optimal solution • But, still fast compared to algorithms • Very sensitive to input parameters – must be configured well to get good results • How large should population be? • How does reproduction occur? • How many generations should be considered? • How much of each generations should be killed off? • Same advantages/disadvantages as simulated annealing

Other Heuristics • Ant colony optimization • Tabu search • Stochastic optimization • Many others

Summary • Optimizations problems search huge solution space for best solution • Many optimization problems are NP-complete • No known efficient solution • Branch and bound helps a little • Heuristics must be used • Find a “good” solution in reasonable time • General optimization problem heuristics • Hill climbing – fast, but gets stuck at local maxima • Simulated annealing – works well but sensitive to parameters • Genetic algorithms - work well but sensitive to parameters • Why should you care about any of this? • Most RC tool problems are NP-complete optimization problems • You now know how to solve almost all tool problems • You can use branch and bound, hill climbing, simulated annealing, genetic algorithms, ant colony optimization, tabu search, etc.

Optimization Problems