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2011 IEEE 26th Annual Conference on Computational Complexity (2011)
San Jose, California USA
June 8, 2011 to June 11, 2011
ISSN: 1093-0159
ISBN: 978-0-7695-4411-3
pp: 23-33
In 1997, H{\aa}stad showed $\NP$-hardness of $(1-\eps, 1/q + \delta)$-approximating $\maxthreelin(\Z_q)$; however it was not until 2007 that Guruswami and Raghavendra were able to show $\NP$-hardness of $(1-\eps, \delta)$-approximating $\maxthreelin(\Z)$. In 2004, Khot--Kindler--Mossel--O'Donnell showed $\UG$-hardness of $(1-\eps, \delta)$-approximating $\maxtwolin(\Z_q)$ for $q = q(\eps,\delta)$ a sufficiently large constant; however achieving the same hardness for $\maxtwolin(\Z)$ was given as an open problem in Raghavendra's 2009 thesis.In this work we show that fairly simple modifications to the proofs of the $\maxthreelin(\Z_q)$ and $\maxtwolin(\Z_q)$ results yield optimal hardness results over $\Z$. In fact, we show a kind of ``bicriteria'' hardness: even when there is a $(1-\eps)$-good solution over $\Z$, it is hard for an algorithm to find a $\delta$-good solution over $\Z$, $\R$, or $\Z_m$ for any $m \geq q(\eps,\delta)$ of the algorithm's choosing.

Y. Wu, Y. Zhou and R. O'Donnell, "Hardness of Max-2Lin and Max-3Lin over Integers, Reals, and Large Cyclic Groups," 2011 IEEE 26th Annual Conference on Computational Complexity(CCC), San Jose, California USA, 2011, pp. 23-33.
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