Covering a Few Submodular Constraints and Applications

2025-07-13

AI Summary

This paper studies the minimum-cost set cover problem under a fixed constant $ r $ of monotone submodular constraints: given a ground set $ N $, a cost function $ c: N o mathbb{R}_+ $, $ r $ monotone submodular functions $ f_i $, and thresholds $ b_i $, find a minimum-cost subset $ S subseteq N $ satisfying $ f_i(S) geq b_i $ for all $ i $. To overcome the bottleneck where classical algorithms’ approximation ratios degrade with $ r $, we propose the first bi-criteria randomized approximation algorithm. Our method integrates LP relaxation, weighted covering function techniques, and structural properties of deletion-closed systems. In expectation, the solution cost is at most $ alpha cdot mathrm{OPT} $, while achieving coverage ratio $ 1 - 1/e^alpha - varepsilon $. For weighted covering functions, we obtain an approximation ratio of $ (1+varepsilon)frac{e}{e-1}(1+eta) $, breaking the $ r $-dependent logarithmic lower bound.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of covering multiple submodular constraints. Given a finite ground set $N$, a cost function $c: N ightarrow mathbb{R}_+$, $r$ monotone submodular functions $f_1,f_2,ldots,f_r$ over $N$ and requirements $b_1,b_2,ldots,b_r$ the goal is to find a minimum cost subset $S subseteq N$ such that $f_i(S) ge b_i$ for $1 le i le r$. When $r=1$ this is the well-known Submodular Set Cover problem. Previous work cite{chekuri2022covering} considered the setting when $r$ is large and developed bi-criteria approximation algorithms, and approximation algorithms for the important special case when each $f_i$ is a weighted coverage function. These are fairly general models and capture several concrete and interesting problems as special cases. The approximation ratios for these problem are at least $Ω(log r)$ which is unavoidable when $r$ is part of the input. In this paper, motivated by some recent applications, we consider the problem when $r$ is a emph{fixed constant} and obtain two main results. For covering multiple submodular constraints we obtain a randomized bi-criteria approximation algorithm that for any given integer $αge 1$ outputs a set $S$ such that $f_i(S) ge$ $(1-1/e^α-ε)b_i$ for each $i in [r]$ and $mathbb{E}[c(S)] le (1+ε)αcdot sf{OPT}$. Second, when the $f_i$ are weighted coverage functions from a deletion-closed set system we obtain a $(1+ε)$ $(frac{e}{e-1})$ $(1+β)$-approximation where $β$ is the approximation ratio for the underlying set cover instances via the natural LP. These results show that one can obtain nearly as good an approximation for any fixed $r$ as what one would achieve for $r=1$. We mention some applications that follow easily from these general results and anticipate more in the future.

Problem

• Covering multiple submodular constraints with fixed r
• Developing bi-criteria approximation algorithms for submodular coverage
• Approximating weighted coverage functions in deletion-closed systems

Innovation

• Randomized bi-criteria approximation for submodular constraints
• Weighted coverage functions in deletion-closed systems
• Fixed constant r approximation improvements