Fast Stochastic Greedy Algorithm for $k$-Submodular Cover Problem

TL;DR

This work tackles the $k$-Submodular Cover problem ($\textsf{kSC}$), a generalization of Submodular Cover to $k$ disjoint sets with a monotone $k$-submodular objective $f:(k+1)^V\to\mathbb{R}_+$ and a threshold $T$, with broad AI applications. It introduces a Fast Stochastic Greedy framework comprising SGOpt, a stochastic greedy routine that uses a guessed optimal size $v$, a truncated objective $f(\cdot)=\min\{f(\cdot),T/2\}$, and random sampling to reduce oracle queries, achieving a bicriteria bound and a controlled solution size. The full FastSG method removes the need for a guessed size by evaluating a geometric sequence of candidate sizes, delivering a $((1+\epsilon)\log(1/\delta),(1-\delta)/2)$-bicriteria approximation with query complexity $O\left(\frac{kn \log^2 n}{\epsilon}\log(n/\delta)\right)$. Empirical results on real networks demonstrate substantial improvements in solution compactness, query efficiency, and runtime compared with state-of-the-art baselines, validating the method's scalability for large-scale AI tasks.

Abstract

We study the $k$-Submodular Cover ($kSC$) problem, a natural generalization of the classical Submodular Cover problem that arises in artificial intelligence and combinatorial optimization tasks such as influence maximization, resource allocation, and sensor placement. Existing algorithms for $\kSC$ often provide weak approximation guarantees or incur prohibitively high query complexity. To overcome these limitations, we propose a \textit{Fast Stochastic Greedy} algorithm that achieves strong bicriteria approximation while substantially lowering query complexity compared to state-of-the-art methods. Our approach dramatically reduces the number of function evaluations, making it highly scalable and practical for large-scale real-world AI applications where efficiency is essential.

Figures (1)

• Figure 1: Performance comparison (function value, query complexity, and size of solution) between FastSG and the state-of-the-art baselines GREEDY and STREAMING.

Theorems & Definitions (9)

• definition thmcounterdefinition: $\textsf{kSC}$ problem
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof