## Abstract

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 independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the bipartite induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the khypergraph 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 independent set problem must run in at least 2n1-ε/r1+ε time. This nearly matches the upper bound of 2^{n/r} [23]. It also improves some hardness results in the domain of parameterized complexity (e.g., [26], [19]). • For any k larger than some constant, there is no polynomial time min {k^{1-ε}, n1/2-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), ̃O ( √ n) } (by Balcan and Blum [3] and an algorithm in this paper). We note an interesting fact that, in contrast to n1/2-ε hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits nδ 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. The proofs of our hardness results rely on 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 induced matching number of dispersers. Then, we show that the n1/2-ε hardness of the last problem follows from nearly tight 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.

Original language | English |
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Title of host publication | Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |

Pages | 370-379 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 2013 |

MoE publication type | A4 Article in a conference publication |

Event | Annual Symposium on Foundations of Computer Science - Berkeley, United States Duration: 27 Oct 2013 → 29 Oct 2013 Conference number: 54 |

### Conference

Conference | Annual Symposium on Foundations of Computer Science |
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Abbreviated title | FOCS |

Country/Territory | United States |

City | Berkeley |

Period | 27/10/2013 → 29/10/2013 |

## Keywords

- Algorithmic pricing
- Approximation algorithms
- Subexponential-time algorithms