An algorithm for $k$-set cover

2026-08-08

AI Summary

This work addresses the exact algorithmic complexity of the $k$-Set Cover problem, where the universe contains $n$ elements and each set has size at most $k$. By leveraging refined combinatorial analysis and advanced techniques in exponential-time algorithm design, the authors develop a new exact algorithm that significantly improves the running time for all sufficiently large $k$. Specifically, they reduce the time complexity from $2^{(1 - 0.929/k)n}$ to $2^{(1 - 1/k + O(1/k^{3/2}))n}$, thereby enhancing the constant factor in the exponent. This result establishes the current best-known exact algorithm for large $k$, surpassing the previous bound by Björklund (STACS 2010).

Abstract

We show that set cover on a universe of size $n$ and with sets of size at most $k$ can be solved in time $2^{(1-1/k+O(1/k^{3/2}))n}$. This improves on a $2^{(1-0.929/k)n}$-time algorithm of Björklund (STACS 2010) for all sufficiently large $k$.

Problem

• k-set cover
• set cover
• exact algorithms
• exponential time algorithms

Innovation

• k-set cover
• exponential-time algorithm
• time complexity improvement
• combinatorial optimization