Hardness of Max-2Lin and Max-3Lin over Integers, Reals, and Large Cyclic Groups

In 1997, Hastad showed NP-hardness of (1 - ε, 1/q + δ)-approximating Max-3Lin(Z
_q
); however it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of (1 - ε, δ)- approximating Max-3Lin(Z). In 2004, Khot-Kindler-Mossel-O'Donnell showed UG-hardness of (1 - ε, δ) approximating Max-2Lin(Z
_q
) for q = q(ε, δ) a sufficiently large constant; however achieving the same hardness for Max-2Lin(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 Max-3Lin(Z
_q
) and Max-2Lin(Z
_q
) results yield optimal hardness results over Z. In fact, we show a kind of "bicriteria" hardness: even when there is a (1 - ε) good solution over Z, it is hard for an algorithm to find a 5-good solution over Z, M, or Z
_m
for any m ≥ q(ε, δ) of the algorithm's choosing.