[Paper Review] Improved Greedy Algorithm for Set Covering Problem

Drona Pratap Chandu|arXiv (Cornell University)|Jun 13, 2015

Computational Geometry and Mesh Generation3 references5 citations

TL;DR

This paper proposes the Big Step Greedy Set Cover (BSGSC) algorithm, an enhanced greedy approach for the NP-hard set covering problem. Unlike the classical greedy algorithm that selects one set at a time, BSGSC selects p sets in each iteration to maximize the number of newly covered elements, significantly improving approximation quality and convergence speed. The method achieves better coverage with fewer iterations, especially when p > 1, and reduces the approximation ratio compared to the standard greedy algorithm.

ABSTRACT

This paper proposes a greedy algorithm named as Big step greedy set cover algorithm to compute approximate minimum set cover. The Big step greedy algorithm, in each step selects p sets such that the union of selected p sets contains greatest number of uncovered elements and adds the selected p sets to partial set cover. The process of adding p sets is repeated until all the elements are covered. When p=1 it behaves like the classical greedy algorithm.

Motivation & Objective

• To address the computational inefficiency and suboptimal approximation ratios of the classical greedy algorithm in solving the set covering problem.
• To reduce the number of iterations required to cover all elements by selecting multiple sets per step instead of one.
• To improve the approximation ratio of the greedy approach by leveraging parallel selection of high-impact sets.
• To provide a scalable and efficient heuristic for large-scale set covering instances.
• To generalize the classical greedy algorithm by introducing a parameter p for batch set selection.

Proposed method

• The algorithm iteratively selects p sets in each step such that their union covers the maximum number of previously uncovered elements.
• The selection of p sets is performed using a greedy heuristic that evaluates all possible combinations of p sets from the current candidate pool.
• After selecting the optimal p-set combination, all p sets are added to the partial set cover, and the covered elements are marked as complete.
• The process repeats until all elements are covered, with p fixed throughout the execution.
• When p = 1, the algorithm reduces to the classical greedy set cover algorithm.
• The method maintains a dynamic list of uncovered elements and updates it after each batch of p sets is added.

Experimental results

Research questions

• RQ1 Can selecting multiple sets per iteration improve the approximation quality of the greedy algorithm for the set covering problem?
• RQ2 How does the performance of the Big Step Greedy Set Cover algorithm compare to the classical greedy algorithm in terms of coverage quality and convergence speed?
• RQ3 What is the optimal value of p that balances computational cost and improvement in approximation ratio?
• RQ4 Does batch selection of p sets reduce the number of iterations required to achieve full coverage?
• RQ5 How does the algorithm scale with increasing problem size and set diversity?

Key findings

• The Big Step Greedy Set Cover algorithm achieves a better approximation ratio than the classical greedy algorithm by selecting p sets per iteration instead of one.
• The algorithm converges faster than the classical greedy approach due to larger coverage gains per iteration.
• When p > 1, the algorithm reduces the number of iterations required to cover all elements, improving efficiency.
• The method maintains the same time complexity order as the classical greedy algorithm but with improved practical performance.
• The algorithm generalizes the classical greedy approach by allowing batch selection, which enhances scalability and solution quality.
• For p = 1, the algorithm reduces to the classical greedy algorithm, validating its correctness as a special case.