AMPL is a high-level algebraic modeling language for mathematical programming. It enables expressing optimization models in a natural mathematical notation close to textbook formulations. AMPL separates model structure from data, allowing the same model to be used with different datasets. Key features: support for linear, nonlinear, integer programming; interfaces with multiple solvers (CPLEX, Gurobi, MINOS, SNOPT); scripting capabilities for scenario analysis; extensive built-in functions. Model structure includes: sets, parameters, variables, objective, constraints. AMPL facilitates rapid prototyping, model maintenance, and allows users to focus on formulation rather than implementation details. Widely used in academia and industry for operations research applications.

Approximation Algorithms provide feasible solutions with guaranteed worst-case performance bounds for NP-hard optimization problems. An α-approximation algorithm for minimization ensures: solution value ≤ α × optimal value. For maximization: solution ≥ (1/α) × optimal. Examples: (1) Christofides algorithm for metric TSP (1.5-approximation); (2) greedy set cover (ln-approximation); (3) vertex cover (2-approximation). Performance ratio analysis balances: solution quality, computational complexity, and practical effectiveness. Some problems admit polynomial-time approximation schemes (PTAS) - algorithms achieving (1+ε)-approximation for any ε > 0. Others are APX-hard, meaning constant-factor approximation is best possible (unless P=NP).

The Assignment Problem optimally assigns n tasks to n agents (one-to-one assignment) minimizing total cost. Formulated as: minimize Σᵢ Σⱼ cᵢⱼxᵢⱼ subject to: Σⱼ xᵢⱼ = 1 for all i (each agent assigned one task), Σᵢ xᵢⱼ = 1 for all j (each task assigned to one agent), xᵢⱼ ∈ {0,1}. Despite being an integer program, it can be solved efficiently in polynomial time using the Hungarian algorithm (O(n³)). Applications include: job assignment, vehicle routing, task scheduling, and resource allocation. Variants include: bottleneck assignment (minimize maximum cost), generalized assignment (agents can handle multiple tasks), and quadratic assignment (costs depend on assignment pairs).