🧬 Genetic Algorithms

1. 🎯 Learning Objectives

• Explain the biological analogy: selection, crossover, mutation
• Trace one generation of a GA: selection → crossover → mutation
• Implement PMX (Partially Mapped Crossover) and Cycle Crossover for TSP
• Analyze GA parameters: population size, crossover rate, mutation rate
• Distinguish path representation vs ordinal representation for TSP

2. 📖 Core Content

3.1 Intuition: Evolution as Search

1. Selection: Fit individuals reproduce more
2. Crossover: Offspring combine traits from two parents
3. Mutation: Random changes introduce variation A Genetic Algorithm (GA) applies these principles to search problems. Individuals are candidate solutions, fitness is solution quality, and evolution drives improvement over generations.

3.2 GA Algorithm Outline

text
GeneticAlgorithm(population_size, generations, fitness, crossover_rate, mutation_rate):
    // Initialize random population
    population = [random_individual() for _ in range(population_size)]
    for gen in range(generations):
        // Evaluate fitness
        fitnesses = [fitness(ind) for ind in population]
        // Selection
        mating_pool = select_parents(population, fitnesses)
        // Crossover
        offspring = []
        for i in range(0, population_size, 2):
            p1, p2 = mating_pool[i], mating_pool[i+1]
            if random() < crossover_rate:
                c1, c2 = crossover(p1, p2)
            else:
                c1, c2 = p1, p2
            offspring.extend([c1, c2])
        // Mutation
        for i in range(population_size):
            if random() < mutation_rate:
                offspring[i] = mutate(offspring[i])
        population = offspring
    return best_individual(population)

3.3 Selection Methods

• Probability of selection ∝ fitness
• $P(i) = f(i) / \sum_j f(j)$ Tournament Selection:
• Pick k individuals randomly
• Select the best among them
• Repeated to fill mating pool Rank Selection:
• Sort by fitness, select proportional to rank
• Reduces domination by super-fit individuals

3.4 Crossover for TSP (Permutation Encoding)

1. Choose two crossover points
2. Copy the segment between points from parent 1 to child 1
3. Map the segment from parent 2 to maintain permutation PMX Example: Parent 1: [1, 2, 3, | 4, 5, 6, | 7, 8, 9] Parent 2: [9, 3, 7, | 8, 2, 5, | 1, 4, 6] Crossover points: positions 3 and 6 Child 1 starts: [?, ?, ?, | 4, 5, 6, | ?, ?, ?] Mapping from parent 2's segment [8, 2, 5]:

• 8 → ? At position 3 in P1 = 4. Put 8 at position where 4 goes? Actually PMX:
  1. Copy segment: Child1[3:6] = [4,5,6]
  2. Map P2[3:6] = [8,2,5] to P1 positions:
     • 8 maps to 4 (from segment mapping)
     • 2 maps to 5
     • 5 maps to 6
  3. Fill remaining positions from P2, resolving via mapping Cycle Crossover (CX):

1. Find cycles in the permutation mapping between parents
2. Copy alternating cycles from each parent Order Crossover (OX):
3. Copy a segment from parent 1
4. Fill remaining positions with cities from parent 2 in order, skipping duplicates

3.5 Mutation for TSP

3.6 Worked Example: GA for TSP (5 cities)

1. [A, B, C, D, E] fitness = 1/distance
2. [A, C, B, D, E]
3. [B, A, C, E, D]
4. [C, D, A, B, E] Fitness calculation: Distance matrix → total tour distance → fitness = 1/total Selection: Roulette wheel based on fitness Crossover (PMX): Combine two selected parents Mutation: Small probability of swapping two cities After one generation, the average fitness should increase.

4. 📐 Key Formulas

5. ⚠️ Common Pitfalls

Pitfall 1: Premature Convergence

Pitfall 2: Invalid Offspring from Crossover

Pitfall 3: Selection Pressure Too High

6. 📝 Practice Questions

Q1: Using PMX, cross Parent1=[1,2,3,4,5,6,7,8,9] and Parent2=[9,3,7,8,2,5,1,4,6] with crossover points 3 and 6.

Answer: Child1: Copy P1[3:6]=[4,5,6]. Mapping from P2[3:6]=[8,2,5] → [4,5,6]: 8↔4, 2↔5, 5↔6. Fill remaining: from P2: 9 (not in child yet) → pos 0; 3 → pos 1; 7 → pos 2; 8 maps to 4 (already used) → pos 6; 1 → pos 7; 4 already in segment → pos 8... Child1: [9,3,7,4,5,6,8,1,2]. Q2: What is the purpose of mutation in a GA?

Answer: Mutation maintains diversity in the population and prevents premature convergence to local optima. It allows the algorithm to explore regions of the search space not reachable through crossover alone. Join Discord PreviousBeam Search & VNDNextACO & Emergent