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$k$-独立图中允许$\frac1k$近似比，在$k$-完美有向图中允许$\frac{1}{2k}$近似比。这些图类是对$k$-退化图、弦图以及区间、伪圆盘等多种几何形状的交图进行泛化的参数化图类。我们将MWIS推广到子模目标函数的情形。给定图$G=(V,E)$和定义在幂集上的非负子模函数$f: 2^V \rightarrow \mathbb{R}_+$，目标为近似求解$\max_{S \in \mathcal{I}_G} f(S)$，其中$\mathcal{I}_G$表示$G$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。第一种方法基于多重线性松弛框架与简单冲突解决机制，得到近似比至少为$\frac{1}{e(k+1)}$的随机算法，该方法同时支持并行（或低自适应度）近似。为实现高效确定性算法，我们受流算法启发，针对归纳$k$-独立图设计了另外两种算法：抢占式贪心算法与原始对偶算法。这些算法不仅更简单快速，且在单调子模情形下，首次为区间、圆盘及伪圆盘交图等先前被研究的特例给出了确定性常数因子近似。