Partially Polynomial Kernels for Set Cover and Test Cover

Abstract

An instance of the $(n-k)$-Set Cover or the $(n-k)$-Test Cover problems is of the form $(\mathcal{U},\mathcal{S},k)$, where $\mathcal{U}$ is a set with $n$ elements, $\mathcal{S}\subseteq 2^\mathcal{U}$ with $|\mathcal{S}|=m$, and $k$ is the parameter. The instance is a Yes-instance of $(n-k)$-Set Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that every element of $\mathcal{U}$ is contained in some set in $\mathcal{S}'$. Similarly, it is a Yes-instance of $(n-k)$-Test Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that for any pair of elements from $\mathcal{U}$, there exists a set in $\mathcal{S}'$ that contains one of them but not the other. It is known in the literature that both $(n-k)$-Set Cover and $(n-k)$-Test Cover do not admit polynomial kernels (under some well-known complexity theoretic assumptions). However, in this paper we show that they do admit “partially polynomial kernels”: we give polynomial time algorithms that take as input an instance $(\mathcal{U},\mathcal{S},k)$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) and return an equivalent instance $(\tilde{\mathcal{U}},\tilde{\mathcal{S}}, \tilde{k})$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) with $\tilde{k} \leq k$ and $| \tilde{\mathcal{U}}|= \mathcal{O}(k^2)$ (respectively, $|\tilde{\mathcal{U}}|=\mathcal{O}(k^7)$). These results allow us to generalize, improve, and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain “sparsity properties.” Using a part of our partial kernelization algorithm for $(n-k)$-Set Cover, we also get an improved fixed-parameter tractable algorithm for this problem which runs in time $\mathcal{O}(4^kk^{\mathcal{O}(1)}(m+n)+mn)$ improving over the previous best of $\mathcal{O}(8^{k+o(k)}(m+n)^{\mathcal{O}(1)})$. On the other hand, the partially polynomial kernel for $(n-k)$-Test Cover gives an algorithm with running time $\mathcal{O}(2^{\mathcal{O}(k^2)}(m+n)^{\mathcal{O}(1)})$. We believe such an approach could also be useful for other covering problems.