Breaking the 1/3 Barrier for constrained $\mathbf{k}$-Submodular Maximization: A Proportional Top-2 Randomized Framework

Abstract

$k$-submodularity generalizes submodularity by allowing each selected element to be assigned one of $k$ labels. We study the maximization of a non-monotone $k$-submodular function, where $k\ge 2$, under matroid and knapsack constraints. We develop a proportional top-2 randomized framework that achieves an approximation ratio of $\sqrt{2}-1\approx 0.4142$ under either constraint, breaking the previous $1/3$ barrier. Once an element is selected, its label is chosen from the two labels with the largest marginal gains, with probabilities proportional to their positive parts. The analysis separates replacement, pure-addition, and residual-deletion losses, and proves a sharp local loss certificate for this rule. We then give a Bellman analysis using the same local certificate. It gives a $1/2$-approximation under a cardinality constraint and strengthens the knapsack guarantee to $1/2$ without changing the algorithm. The $1/2$ guarantees are oracle-optimal for $k=2$ and asymptotically tight as $k\to \infty$. The value-oracle complexities are $O(Bnk)$ for cardinality, where $B$ is the cardinality bound, $O(n^{2}k)$ for matroid, and $O(n^3{k}^2)$ for knapsack.