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 $(U,f)$ be a multiset built from set $U=\{e_1,e_2,\dots ,e_m\}$ and function $f:U\to N^{⁎}$. ESM is defined as follows: given $(U,f)$ and a collection $S=\{S_1,S_2,\dots ,S_n\}$ of n subsets of $U$, is it possible to find a multiset $(S^{\prime },g)$ with $S^{\prime }=\{S_1^{\prime },S_2^{\prime },\dots ,S_n^{\prime }\}$ and $g:S^{\prime }\to N$, such that (i) $S_i^{\prime }\subseteq S_i$ for every $1\le i\le n$, and (ii) each element of $U$ appears as many times in $(U,f)$ as in $(S^{\prime },g)$? 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 (EESM), which asks that each element of $U$ appears in exactly one subset $S_i^{\prime }$ of $S^{\prime }$; (ii) Maximum Exclusive Exact Subset MultiCover (Max-EESM), an optimization version of EESM, which asks that a maximum number of elements of $U$ appear in exactly one subset $S_i^{\prime }$ of $S^{\prime }$. For both variants, we provide several complexity results; in particular we present a 2-approximation algorithm for Max-EESM, that we prove to be tight. For these three problems, we also provide an Integer Linear Programming (ILP) formulation.