Minimum set cover with incompatible sets

Question

I'm interested in a variant of minimum set cover where some sets are ``incompatible'' (they can't be chosen simultaneously).

To state it more formally: We have a finite base set $X$ and a family $\mathcal{R}$ of subsets of $X$. We also have an undirected graph $G$ with vertex set $\mathcal{R}$ and edges representing incompatibilities between sets. The goal is to find a minimum set cover of $X$ which is also an independent set of $G$.

Does this problem have a name? Or is it a special case of a studied problem? Or can it be reduced to a well-studied problem with a small blowup in runtime? (by "small blowup" I mean not simply polynomial, but preferably a small degree polynomial).

I'm particularly interested in the geometric variant where $X$ is a set of points and the sets in $\mathcal{R}$ are defined as the intersection of $X$ with some simple geometric ranges (in which case tools such as VC-dimension, $\epsilon$-nets, range-searching and so on might come in handy for approximation algorithms). But I welcome any relevant reference.

Answer 1

Your problem can be stated as a minimum weight maximal independent set problem.

Construction:

Construct a bipartite graph $G = (L,R,E)$, where the right partition $R$ corresponds to $\mathcal{R}$ and the left partition $L$ corresponds to $X$. An edge $(u,v) \in E$ iff an element $u$ is contained in the set $v$.

Let $G'$ be the updated graph obtained by adding edges within $R$ based on the incompatibility of vertices in $R$.

Furthermore, assign a weight of $\infty$ on each vertex of $L$ and weight $1$ on each vertex of $R$. Also, attach a new pendant vertex of weight $0$ to each vertex in $R$. Let $G_{w}$ be this new graph.

Now, your problem can be stated as finding minimum weight maximal independent set problem (MMIS) on $G_{w}$. It is a standard problem discussed here. Your problem has a feasible solution if and only if MMIS has a finite value. Also, your problem has a solution of size $k$ if and only if MMIS has a value $k$.

Correctness:

Suppose, a feasible instance of the independent set cover problem is $S_{1}, \dotsc, S_{k}$. Then, the maximal independent set contains vertices in $R$ corresponding to $S_{1}, \dotsc, S_{k}$ and pendant vertices corresponding to the remaining sets. It is easy to see that it is an independent set and it is also maximal. Moreover, the weight of the independent set is exactly $k$.

Similarly, you can prove the other direction, i.e., if there is a maximal independent set of value $k$, there is an independent set cover of size $k$.

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Note: Instead of assigning $\infty$ value to the vertices in $L$, you can also assign some sufficiently large finite value, say $|\mathcal{R}|+1$. That will also work.

Answer 2

There are two recent papers that may be of interest in this context:

• This paper studies exactly the problem you described: finding set covers whose sets are independent in some additionally given graph: https://doi.org/10.1007/s00224-020-10022-9

• This paper studies set cover variants where the sets have to be independent in one or more given matroids: http://doi.org/10.1016/j.dam.2021.03.014