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

2026-09-14

AI Summary

本文解决了非单调k-次模函数在拟阵和背包约束下的最大化问题，通过比例Top-2随机化框架突破了1/3的近似比障碍。

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.

Problem

• non-monotone k-submodular maximization
• matroid constraint
• knapsack constraint

Innovation

• proportional top-2 randomized framework
• non-monotone k-submodular maximization
• matroid and knapsack constraints
• approximation ratio