Set Cover is one of the well-known classical NP-hard problems. We study the conflict-free version of the Set Cover problem. Here we have a universe \(\mathcal {U}\), a family \(\mathcal {F}\) of subsets of \(\mathcal {U}\) and a graph \(G_{\mathcal {F}}\) on the vertex set \(\mathcal {F}\) and we look for a subfamily \(\mathcal {F}^{\prime } \subseteq \mathcal {F}\) of minimum size that covers \(\mathcal {U}\) and also forms an independent set in \(G_{\mathcal {F}}\). We study conflict-free Set Cover in parameterized complexity by restricting the focus to the variants where Set Cover is fixed parameter tractable (FPT). We give upper bounds and lower bounds for the running time of conflict-free version of Set Cover with and without duplicate sets along with restrictions to the graph classes of \(G_{\mathcal {F}}\). For example, when pairs of sets in \(\mathcal {F}\) intersect in at most one element, for a solution of size k, we give

• an \(f(k)|\mathcal {F}|^{o(k)}\) lower bound for any computable function f assuming ETH even if \(G_{\mathcal {F}}\) is bipartite, but

• an \(O^{*}(3^{k^{2}})\) FPT algorithm (\(\mathcal {O}^{*}\) notation ignores polynomial factors of input) when \(G_{\mathcal {F}}\) is chordal.

Set Cover是著名的经典NP难题之一。我们研究了Set Cover问题的无冲突版本。这里我们有一个宇宙\（\ mathcal {U} \），一个由\（\ mathcal {U} \）的子集组成的族\（\ mathcal {F } \）和一个图\（G _ {\ mathcal {F} } \）在顶点集\（\ mathcal {F} \）上，我们寻找一个最小子集\（\ mathcal {F} ^ {\ prime} \ subseteq \ mathcal {F} \）覆盖\（ \ mathcal {U} \）并在\（G _ {\ mathcal {F}} \）中形成一个独立的集合。我们研究无冲突套装通过将焦点集中在Set Cover是固定参数可处理（FPT）的变量上，从而降低了参数化的复杂性。我们给出了有无重复集的Set Cover无冲突版本的运行时间的上限和下限，以及对\（G _ {\ mathcal {F}} \）图类的限制。例如，当\（\ mathcal {F} \）中的集合对相交至多一个元素时，对于大小为k的解，我们给出

• 即使假设\（G _ {\ mathcal {F}} \）是二分的，但假设ETH的任何可计算函数f的\（f（k）| \ mathcal {F} | ^ {o（k）} \）下限

• 一个\（O ^ {*}（3 ^ {K ^ {2}}）\） FPT算法（\（\ mathcal {ö} ^ {*} \）表示法忽略输入的多项式因子）时\（G _ {\ mathcal {F}} \）是和弦的。