← 研踪 PaperTrailarXiv:2609.19623 · v2
Near-Logarithmic Inapproximability of Parameterized Set Cover
首次提交 2026-09-17 11:15
版本更新 2026-11-03 17:06
雷达观测 2026-11-04 23:39
摘要
In the $γ$-gap $k$-SetCover problem, we are given a family $\mathcal{S}$ over a universe $U$, and the goal is to distinguish between the case where $\mathcal{S}$ contains $k$ sets covering $U$, and the case where at least $γk$ sets in $\mathcal{S}$ are needed to cover $U$. We show that, for every $k^{Ω(k^2)}\leρ\le n^{O(1)}$, it is W[1]-hard to approximate $k$-SetCover within ratio $Ω\big(\frac{\logρ}{k^2\log\logρ}\big)$ on instances with $n$ sets and universe size at most $ρ\log n$. We also show that under the randomized $k$-OV Hypothesis, there is no bounded-error randomized $f(k)n^{o(k)}$-time $o\big(\frac{\log N}{\log\log N}\big)$-approximation algorithm for $k$-SetCover. These results improve on the previous $(\log n)^{1/O(k)}$-inapproximability. For fixed $k$, this matches the polynomial-time $(\lnρ+O(1))$-approximation up to a $\log\logρ$ factor. We further establish a connection between small universes and large approximation gaps. We show that for every $1\leγ\le\min\big\{\frac{\log u}{\log\log N},\frac{\log u}{k^2\log\log u}\big\}$, exact $k$-SetCover with universe size $u^{Θ(1/γ)}$ is equivalent to $γ$-gap $k$-SetCover with universe size $u$ under randomized Turing reductions. This suggests a barrier to proving superconstant-gap W[2]-hardness: such a result would also make exact $k$-SetCover with universe size $n^{o(1)}$ W[2]-hard under randomized FPT Turing reductions. Finally, we analyze the VC dimensions of the set systems produced by our reduction. Write $d$ and $d^*$ for the primal and dual VC dimensions respectively. We show that for $γ=o(\frac{\log N}{\log\log N})$, $γ$-gap $k$-SetCover is W[1]-hard even when $d=O(k^{2}γ\log(kγ))$ and $d^*=O(kγ)$. The bound for $d^*$ matches the $O(d^*\log k)$-approximation algorithm up to a $k\log k$ factor.
观测到的版本
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v2 · 2026-11-03 17:06 · 最新观测版本
Near-Logarithmic Inapproximability of Parameterized Set Cover
In the $γ$-gap $k$-SetCover problem, we are given a family $\mathcal{S}$ over a universe $U$, and the goal is to distinguish between the case where $\mathcal{S}$ contains $k$ sets covering $U$, and the case where at least $γk$ sets in $\mathcal{S}$ are needed to cover $U$. We show that, for every $k^{Ω(k^2)}\leρ\le n^{O(1)}$, it is W[1]-hard to approximate $k$-SetCover within ratio $Ω\big(\frac{\logρ}{k^2\log\logρ}\big)$ on instances with $n$ sets and universe size at most $ρ\log n$. We also show that under the randomized $k$-OV Hypothesis, there is no bounded-error randomized $f(k)n^{o(k)}$-time $o\big(\frac{\log N}{\log\log N}\big)$-approximation algorithm for $k$-SetCover. These results improve on the previous $(\log n)^{1/O(k)}$-inapproximability. For fixed $k$, this matches the polynomial-time $(\lnρ+O(1))$-approximation up to a $\log\logρ$ factor. We further establish a connection between small universes and large approximation gaps. We show that for every $1\leγ\le\min\big\{\frac{\log u}{\log\log N},\frac{\log u}{k^2\log\log u}\big\}$, exact $k$-SetCover with universe size $u^{Θ(1/γ)}$ is equivalent to $γ$-gap $k$-SetCover with universe size $u$ under randomized Turing reductions. This suggests a barrier to proving superconstant-gap W[2]-hardness: such a result would also make exact $k$-SetCover with universe size $n^{o(1)}$ W[2]-hard under randomized FPT Turing reductions. Finally, we analyze the VC dimensions of the set systems produced by our reduction. Write $d$ and $d^*$ for the primal and dual VC dimensions respectively. We show that for $γ=o(\frac{\log N}{\log\log N})$, $γ$-gap $k$-SetCover is W[1]-hard even when $d=O(k^{2}γ\log(kγ))$ and $d^*=O(kγ)$. The bound for $d^*$ matches the $O(d^*\log k)$-approximation algorithm up to a $k\log k$ factor.
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