Approximation algorithm for the partial set multi-cover problem

Abstract

Partial set cover problem and set multi-cover problem are two generalizations of the set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets $\mathcal S\subseteq 2^E$, a total covering ratio q, each set $S\in \mathcal S$ is associated with a cost $c_S$, each element $e\in E$ is associated with a covering requirement $r_e$, the goal is to find a minimum cost sub-collection ${\mathcal {S}}'\subseteq {\mathcal {S}}$ to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least $r_e$ sets of ${\mathcal {S}}'$. Denote by $r_{\max }=\max \{r_e:e\in E\}$ the maximum covering requirement. We present an $(O (r_{\max }\log ^2n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q})),1-\varepsilon )$-bicriteria approximation algorithm, that is, the output of our algorithm has cost $O(r_{\max }\log ^2 n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q}))$ times of the optimal value while the number of fully covered elements is at least $(1-\varepsilon )q|E|$.

Appendix: A flaw in the conference version

A preliminary version of this paper was presented in INFOCOM2017. Making using of an $\alpha$-approximation algorithm for MDSC, it was claimed that one can obtain an $\alpha H\lceil qn\rceil$-approximation algorithm for PSMC. However, there is a flaw. The algorithm in that paper greedily selects densest sub-collections until at least qn elements are fully covered. Then it prunes the last sub-collection ${\mathcal {R}}$ by greedily selecting sub-collections of ${\mathcal {R}}$ consisting of at most $r_{\max }$ sets until the covering requirement is satisfied. Suppose the sub-collections obtained in the pruning step are $\mathcal R_1',\ldots ,{\mathcal {R}}_l'$. The approximation analysis relies on the following inequality:

$$\begin{aligned} \frac{c({\mathcal {R}}_1')}{|{\mathcal {C}}({\mathcal {R}}_1')|}\le \frac{c({\mathcal {R}}_2')}{|{\mathcal {C}}({\mathcal {R}}_2')|}\le \cdots \le \frac{c({\mathcal {R}}_l')}{|{\mathcal {C}}({\mathcal {R}}_l')|}. \end{aligned}$$

(13)

However, this is not true. Consider the following example

Example 5.1

${\mathcal {S}}=\{S_1,S_2,S_3\}$ with $S_1=\{e_1,e_2\}$, $S_2=\{e_1,e_3\}$, $S_3=\{e_2,e_3\}$, $r(e_1)=r(e_2)=r(e_3)=r_{\max }=2$, $c(S_1)=c(S_2)=c(S_3)=1$, and $q=2/3$.

For this example, the densest sub-collection of ${\mathcal {S}}$ is ${\mathcal {R}}={\mathcal {S}}$. Then the pruning step selects $\mathcal R_1'=\{S_1,S_2\}$ and ${\mathcal {R}}_2'=\{S_3\}$ sequentially. Notice that

$$\begin{aligned} \frac{c({\mathcal {R}}_1')}{|{\mathcal {C}}({\mathcal {R}}_1')|}=2>\frac{1}{2}=\frac{c({\mathcal {R}}_2')}{|{\mathcal {C}}({\mathcal {R}}_2')|}. \end{aligned}$$

The reason why inequality (13) does not hold is because $|{\mathcal {C}}({\mathcal {R}}')|$ is not a submodular function, and it is difficult to bypass this obstacle. Obtaining an approximation algorithm achieving a guaranteed performance ratio in the classic sense is a very challenging problem.