核心概念
A simple and efficient greedy algorithm that finds a 𝑘-crashing plan with an approximation ratio of 1/(1 + 1/2 + ... + 1/𝑘).
要約
The content presents a problem in project management called the 𝑘-crashing problem, where the goal is to find the minimum cost to speed up a project by 𝑘 days. The authors propose a simple greedy algorithm and analyze its approximation performance.
Key highlights:
- The project is modeled as an activity-on-edge network (AOE network), where each job/activity is represented as an edge. Some jobs must be finished before others can start, as described by the network topology.
- To speed up the project, the manager can "crash" a few jobs by investing extra resources, but each job has a lower bound on its duration due to technological limits.
- The greedy algorithm works by iteratively finding the minimum cost way to shorten the project duration by 1 day, and repeating this process 𝑘 times.
- The authors prove that this greedy algorithm achieves an approximation ratio of 1/(1 + 1/2 + ... + 1/𝑘), meaning the total cost of the solution is at most this factor times the optimal cost.
- The authors also analyze a related problem called 𝑘-LIS (finding 𝑘 disjoint increasing subsequences of maximum total length) and show a (1-1/e)-approximation algorithm for it.
- The proofs rely on a careful decomposition of the optimal solution and analysis of the properties of the critical graph of the project network.