## Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the {\em subexponential-time} inapproximability of the {\em maximum independent set} problem, a question studied in the area of {\em parameterized complexity}. The second is the hardness of approximating the {\em maximum induced matching} on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the {\em $k$-hypergraph pricing} problem, a fundamental problem arising from the area of {\em algorithmic game theory}. In particular, assuming the Exponential Time Hypothesis, our two main results are:

• For any $r$ larger than some constant, any $r$-approximation algorithm for the maximum independent set problem must run in time at least $2^{n^{1-\epsilon}/r^{1+\epsilon}}$. This nearly matches the upper bound of $2^{n/r}$ [Cygan et al:  Manuscript 2008]. It also improves some hardness results in the domain of parameterized complexity (e.g. [Escoffier et al: Manuscript 2012] and [Chitnis et al: Manuscript 2013]).
• For any $k$ larger than some constant, there is no polynomial time $\min \{k^{1-\epsilon}, n^{1/2-\epsilon}\}$-approximation algorithm for the $k$-hypergraph pricing problem , where $n$ is the number of vertices in an input graph. This almost matches the upper bound of  $\min \{O(k), \tilde O(\sqrt{n})\}$  (by Balcan and Blum in [Balcan and Blum: Theory of Computing 2007] and an algorithm in this paper).

We note an interesting fact that, in contrast to $n^{1/2-\epsilon}$ hardness, the $k$-hypergraph pricing problem admits $n^{\delta}$ approximation for any $\delta >0$ in quasi-polynomial time. This puts pricing problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.
The proofs of our hardness results rely on several unexpectedly tight connections between the three problems. First, we establish a connection between the first and second problems by proving a new graph-theoretic property related to an {\em induced matching number} of dispersers. Then, we show that the $n^{1/2-\epsilon}$ hardness of the last problem follows from nearly tight {\em subexponential time} inapproximability of the first problem, illustrating a rare application of the second type of inapproximability result to the first one. Finally, to prove the subexponential-time inapproximability of the first problem, we construct a new PCP with several properties; it is sparse and has nearly-linear size, large degree, and small free-bit complexity. Our PCP requires no ground-breaking ideas but rather a very careful assembly of the existing ingredients in the PCP literature.