Algorithms for covering multiple submodular constraints and applications

Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set N , weight function $$w: \rightarrow \mathbb {R}_+$$ w : N → R + r monotone functions $$f_1,f_2,\ldots ,f_r$$ f 1 , 2 … r over and requirements $$k_1,k_2,\ldots ,k_r$$ k goal is to find minimum subset $$S \subseteq N$$ S ⊆ such that $$f_i(S) \ge k_i$$ i ( ) ≥ for $$1 \le i r$$ ≤ . refer this as Multi-Submod-Cover it was recently considered by Har-Peled Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260 2018) who were motivated an application in geometry. Even with $$r=1$$ = generalizes well-known Submodular Set Cover ( Submod-SC ), can also be easily reduced A simple greedy algorithm gives $$O(\log (kr))$$ O log approximation where $$k = \sum _i ∑ ratio cannot improved general case. In paper, several concrete applications, we two ways improve upon given algorithm. First, give bicriteria covers each constraint within factor $$(1-1/e-\varepsilon )$$ - / e ε while incurring $$O(\frac{1}{\epsilon }\log r)$$ ϵ cost. Second, special case when $$f_i$$ obtained from truncated coverage obtain previous work on partial cover Partial-SC integer programs CIPs ) vertex constraints Bera et al. (Theoret Comput Sci 555:2–8 2014). Both these algorithms are based mathematical programming relaxations avoid limitations demonstrate implications our related ideas applications ranging geometric problems clustering outliers. Our highlights utility high-level model lens submodularity addressing class problems.