We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halld\'orsson [SODA 1995] showed that for every $\delta>0$ IS has a polynomial-time $(\frac{d-1}{2}+\delta)$-approximation in $K_{1,d}$-free graphs. We extend this result by showing that $K_{a,b}$-free graphs admit a polynomial-time $O(\alpha(G)^{1-1/a})$-approximation, where $\alpha(G)$ is the size of a maximum independent set in $G$. Furthermore, we complement the result of Halld\'orsson by showing that for some $\gamma=\Theta(d/\log d),$ there is no polynomial-time $\gamma$-approximation for these graphs, unless NP = ZPP. Bonnet et al. [IPEC 2018] showed that IS parameterized by the size $k$ of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least $4$, (2) the star $K_{1,4}$, and (3) any tree with two vertices of degree at least $3$ at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken condition (2) that $G$ does not contain $K_{1,5}$). First, under the ETH, there is no $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Then, under the deterministic Gap-ETH, there is a constant $\delta>0$ such that no $\delta$-approximation can be computed in $f(k) \cdot n^{O(1)}$ time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime $f(k)\cdot n^{o(k)}$. Finally, we consider the parameterization by the excluded graph $H$, and show that under the ETH, IS has no $n^{o(\alpha(H))}$ algorithm in $H$-free graphs and under Gap-ETH there is no $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$.

中文翻译：

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无 $H$ 图中独立集的参数化不可逼近性

我们研究无$H$ 图中的独立集（IS）问题，即不包括某些固定图$H$ 作为诱导子图的图。我们证明了多项式时间和参数化算法的几个不可逼近性结果。Halld\'orsson [SODA 1995] 表明，对于每一个 $\delta>0$，IS 在 $K_{1,d 中都有一个多项式时间 $(\frac{d-1}{2}+\delta)$-approximation }$-free 图表。我们通过证明 $K_{a,b}$-free 图承认多项式时间 $O(\alpha(G)^{1-1/a})$-approximation 来扩展这个结果，其中 $\alpha(G )$ 是 $G$ 中最大独立集的大小。此外，我们通过证明对于某些 $\gamma=\Theta(d/\log d),$ 不存在这些图的多项式时间 $\gamma$-近似值来补充 Halld\'orsson 的结果，除非 NP = ZPP。邦内特等人。[IPEC 2018] 表明，由独立集的大小 $k$ 参数化的 IS 在不包含（1）恒定长度至少 $4$ 的循环，（2）星 $ 的图上是 W[1]-hard K_{1,4}$，以及 (3) 任何具有两个度数至少为 $3$ 的顶点且距离不变的树。我们通过在几乎同一类图的不同复杂性假设下证明三个不近似性结果来加强这个结果（我们削弱了 $G$ 不包含 $K_{1,5}$ 的条件（2））。首先，在ETH下，没有任何可计算函数$f$的$f(k)\cdot n^{o(k/\log k)}$算法。然后，在确定性 Gap-ETH 下，存在一个常数 $\delta>0$，使得在 $f(k) \cdot n^{O(1)}$ 时间内无法计算 $\delta$-近似值。还，在更强的随机化 Gap-ETH 下，没有运行时 $f(k)\cdot n^{o(k)}$ 的近似算法。最后，我们考虑排除图$H$的参数化，并表明在ETH下，IS在$H$-free图和Gap-下没有$n^{o(\alpha(H))}$算法ETH 没有 $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$。