Maximum weight independent set (MWIS) admits a $\frac1k$-approximation in inductively $k$-independent graphs and a $\frac{1}{2k}$-approximation in $k$-perfectly orientable graphs. These are a a parameterized class of graphs that generalize $k$-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph $G=(V,E)$ and a non-negative submodular function $f: 2^V \rightarrow \mathbb{R}_+$, the goal is to approximately solve $\max_{S \in \mathcal{I}_G} f(S)$ where $\mathcal{I}_G$ is the set of independent sets of $G$. We obtain an $\Omega(\frac1k)$-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least $\frac{1}{e(k+1)}$. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively $k$-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

翻译：最大权独立集（MWIS）在归纳k-独立图中具有$\frac1k$近似比，在k-完美有向图中具有$\frac{1}{2k}$近似比。这些图类是一类参数化图，推广了k-退化图、弦图以及各种几何形状（如区间、伪圆盘等）的交图。我们考虑将MWIS推广到子模目标函数。给定图$G=(V,E)$和非负子模函数$f: 2^V \rightarrow \mathbb{R}_+$，目标是近似求解$\max_{S \in \mathcal{I}_G} f(S)$，其中$\mathcal{I}_G$是$G$的独立集族。我们在上述两类图中获得了该问题的$\Omega(\frac1k)$近似比。第一种方法基于多线性松弛框架和简单冲突解决机制，由此得到近似比至少为$\frac{1}{e(k+1)}$的随机算法。该方法还能产生并行（或低适应性）近似。受设计高效确定性算法目标的驱动，我们描述了针对归纳k-独立图的另外两种算法，这些算法受流算法研究启发：抢占式贪婪算法和原始-对偶算法。除了更简单和更快外，这些算法在单调子模情形下，为先前考虑过的各种特例（如区间、圆盘和伪圆盘的交图）首次提供了确定性常数因子近似。