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 subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem
... arising from the area of 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 at least 2^n^1-ϵ/r^1+ϵ time. This nearly matches the upper bound of 2^n/r (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^1-ϵ, n^1/2-ϵ)-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), Õ(√(n))) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^1/2-ϵ hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^δ approximation for any δ >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.