The Exact Subset MultiCover Problem

Abstract

In this paper, we study the Exact Subset MultiCover problem (or ESM), which can be seen as an extension of the well-known Set Cover problem. Let $({\mathcal {U},f})$ be a multiset built from set $\mathcal {U}{=\{e_1,e_2,\dots ,e_m\}}$ and function $f: \mathcal {U} \rightarrow \mathbb {N}^*$. ESM is defined as follows: given $({\mathcal {U},f})$ and a collection $\mathcal {S} = {\{S_1,S_2,\dots ,S_n\}}$ of n subsets of $\mathcal {U}$, is it possible to find a multiset $({\mathcal {S'},g})$ with $\mathcal {S}'={\{S'_1,S'_2,\dots ,S'_n\}}$ and $g: \mathcal {\mathcal {S}'} \rightarrow \mathbb {N}$, such that (i) $S'_i \subseteq S_i$ for every $1\le i \le n$, and (ii) each element of $\mathcal {U}$ appears as many times in $(\mathcal {U}, f)$ as in $({\mathcal {S'},g})$ ? We study this problem under an algorithmic viewpoint and provide diverse complexity results such as polynomial cases, NP-hardness proofs and FPT algorithms. We also study two variants of ESM: (i) Exclusive Exact Subset MultiCover (E ESM), which asks that each element of $\mathcal {U}$ appears in exactly one subset $S'_i$ of $\mathcal {S}'$; (ii) Maximum Exclusive Exact Subset MultiCover (Max-E ESM), an optimisation version of E ESM, which asks that a maximum number of elements of $\mathcal {U}$ appear in exactly one subset $S'_i$ of $\mathcal {S}'$. For both variants, we provide several complexity results; in particular we present a 2-approximation algorithm for Max-E ESM, that we prove to be tight.

E. Benoist et al.—supported by the French National Research Agency (ANR-18-CE45-004), ANR DeepProt.