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

—In 1997, H˚astad showed NP-hardness of (1 − , 1/q + δ)-approximating Max-3Lin(Zq); however it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of (1 − , δ)approximating Max-3Lin(Z). In 2004, Khot–Kindler– Mossel–O’Donnell showed UG-hardness of (1 − , δ)approximating Max-2Lin(Zq) for q = q( , 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(Zq) and Max-2Lin(Zq) results yield optimal hardness results over Z. In fact, we show a kind of “bicriteria” hardness: even when there is a (1− )good solution over Z, it is hard for an algorithm to find a δ-good solution over Z, R, or Zm for any m ≥ q( , δ) of the algorithm’s choosing.

• Type: Journal
• Year: 2011
• Where: COCO
• Authors: Ryan O'Donnell, Yi Wu, Yuan Zhou
• Keywords: Algorithms, COCO 2011, Khot, Raghavendra, Zq