Breaking the 1/3 Barrier for $\boldsymbol{k}$-Submodular Maximization under Matroid and Knapsack Constraints: A Proportional Top-2 Randomized Framework

Abstract

$k$-submodularity generalizes submodularity by allowing each selected element to be assigned one of $k$ labels, rather than being merely selected or not selected. We study the problem of maximizing a nonnegative non-monotone $k$-submodular function, where $k\ge 2$, under classical support constraints, including a single matroid constraint and a single knapsack constraint. Previously, the best known approximation guarantees for non-monotone constrained $k$-submodular maximization had long remained at $1/3$ or $1/3-\varepsilon$, even in basic settings such as cardinality, matroid, and knapsack constraints. We show that this $1/3$ barrier is not inherent: for both the matroid and knapsack settings considered here, we give randomized polynomial-time algorithms achieving an approximation ratio of $\sqrt{2}-1\approx 0.4142$. The algorithms use a simple randomized greedy rule: once an element is selected, its label is chosen only from the two labels with the largest marginal gains, with probabilities proportional to the positive parts of these two gains. The value-oracle query complexity is $O(n^2k)$ in the matroid setting and $O(n^3k^2)$ in the knapsack setting. These results give the first approximation guarantees exceeding $1/3$ for non-monotone $k$-submodular maximization under matroid and knapsack constraints.