Greedy Set Cover Estimations

More precise estimation of the greedy algorithm complexity for a special case of the set cover problem is given in this paper.

💡 Research Summary

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The paper addresses the classic Set Cover problem, which is known to be NP‑complete, and focuses on improving the theoretical bound for the greedy heuristic when the input matrix has a particular density property. In the standard formulation, a finite ground set A with |A| = n and a family F of m subsets are represented by an n × m binary matrix where rows correspond to subsets and columns to elements. The goal is to select the smallest sub‑family F_C ⊆ F that covers every column at least once.

The greedy algorithm repeatedly picks the row that covers the largest number of still‑uncovered columns. In the worst case, this algorithm is known to achieve an approximation factor of H_n ≈ ln n + O(1). However, this bound is derived for arbitrary instances and can be overly pessimistic for matrices that are “dense” in a specific sense.

The authors consider a special case: each column contains at least m·γ ones, where 0 < γ ≤ 1 is a constant that measures the minimum column density. Under this assumption the total number of ones in the matrix is at least m·n·γ, and consequently there exists at least one row containing at least n·γ ones. The paper builds on a previously known inequality (cited from reference