Restricted Parameter Range Promise Set Cover Problems Are Easy

Let $({\bf U},{\bf S},d)$ be an instance of Set Cover Problem, where ${\bf U}={u_1,…,u_n}$ is a $n$ element ground set, ${\bf S}={S_1,…,S_m}$ is a set of $m$ subsets of ${\bf U}$ satisfying $\bigcup_{i=1}^m S_i={\bf U}$ and $d$ is a positive integer. In STOC 1993 M. Bellare, S. Goldwasser, C. Lund and A. Russell proved the NP-hardness to distinguish the following two cases of ${\bf GapSetCover_{\eta}}$ for any constant $\eta > 1$. The Yes case is the instance for which there is an exact cover of size $d$ and the No case is the instance for which any cover of ${\bf U}$ from ${\bf S}$ has size at least $\eta d$. This was improved by R. Raz and S. Safra in STOC 1997 about the NP-hardness for ${\bf GapSetCover}_{clogm}$ for some constant $c$. In this paper we prove that restricted parameter range subproblem is easy. For any given function of $n$ satisfying $\eta(n) \geq 1$, we give a polynomial time algorithm not depending on $\eta(n)$ to distinguish between {\bf YES:} The instance $({\bf U},{\bf S}, d)$ where $d>\frac{2 |{\bf S}|}{3\eta(n)-1}$, for which there exists an exact cover of size at most $d$; {\bf NO:} The instance $({\bf U},{\bf S}, d)$ where $d>\frac{2 |{\bf S}|}{3\eta(n)-1}$, for which any cover from ${\bf S}$ has size larger than $\eta(n) d$. The polynomial time reduction of this restricted parameter range set cover problem is constructed by using the lattice.

💡 Research Summary

The paper revisits the classic “promise” version of the Set Cover problem, known as GapSetCover(_\eta), where one must distinguish between instances that admit an exact cover of size (d) (YES) and instances where any cover requires at least (\eta d) sets (NO). Earlier works (Bellare, Goldwasser, Lund, Russell 1993; Raz, Safra 1997) proved that this distinction is NP‑hard for any constant (\eta>1) and even for (\eta = c\log m). The author focuses on a restricted parameter range: instances where the cover size parameter (d) satisfies

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