Greedy Algorithm

The greedy algorithm is an optimization strategy that makes locally optimal choices at each step, with the core principle of gradually approaching a global optimum through successive local optimizations. Although this method does not guarantee a global optimum in all cases, it is widely favored in practical applications due to its efficiency and simplicity.

In 0-1 integer programming problems, variables are restricted to values of 0 or 1. Such problems are commonly applied in resource allocation, scheduling, and network design. The greedy algorithm can efficiently solve 0-1 integer programming problems using implicit enumeration, which progressively determines variable values while avoiding complete enumeration of all possible combinations, thereby significantly reducing computational complexity.

When implementing implicit enumeration in MATLAB to solve 0-1 integer programming problems, the first step involves defining the objective function and constraints. A greedy strategy can determine variable prioritization—for instance, selecting variables for assignment based on the magnitude of their coefficients in the objective function. During each iteration, the algorithm fixes values for certain variables following the direction of the current best solution and recursively or iteratively processes the remaining variables. MATLAB's vectorization capabilities and Optimization Toolbox functions like intlinprog can further enhance computational efficiency, making the approach suitable for medium-scale problems.

Although the greedy algorithm may not ensure a global optimum in 0-1 integer programming, it can rapidly yield high-quality approximate solutions for problems with specific structures, such as those exhibiting submodularity or satisfying the greedy choice property. For more complex 0-1 integer programming challenges, combining the greedy algorithm with other optimization techniques—such as branch and bound or dynamic programming—can improve solution accuracy and broaden applicability.