AI 中文总结
针对非单调 $k$-子模最大化在拟阵和背包约束下的问题，提出比例 Top-2 随机化贪心算法，将近似比从 1/3 提升至 $\sqrt{2}-1\approx 0.4142$，首次突破 1/3 障碍。
AI 中文摘要
$k$-子模性是子模性的推广，它允许每个被选中的元素被分配 $k$ 个标签之一，而不是仅仅被选中或未被选中。我们研究了在经典支持约束（包括单一拟阵约束和单一背包约束）下最大化非负非单调 $k$-子模函数的问题，其中 $k\ge 2$。此前，非单调约束 $k$-子模最大化的最佳已知近似保证长期停留在 $1/3$ 或 $1/3-\varepsilon$，即使在基数、拟阵和背包约束等基本设置中也是如此。我们证明这一 $1/3$ 障碍并非固有：对于这里考虑的拟阵和背包两种设置，我们给出了随机多项式时间算法，实现了 $\sqrt{2}-1\approx 0.4142$ 的近似比。这些算法使用一种简单的随机贪心规则：一旦一个元素被选中，其标签仅从具有最大边际增益的两个标签中选择，选择概率与这两个增益的正部分成比例。在拟阵设置中，值预言查询复杂度为 $O(n^2k)$，在背包设置中为 $O(n^3k^2)$。这些结果首次给出了在拟阵和背包约束下非单调 $k$-子模最大化超过 $1/3$ 的近似保证。
英文摘要
$k$-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.
