TY - GEN

T1 - Hardness of Max-2Lin and Max-3Lin over integers, reals, and large cyclic groups

AU - O'Donnell, Ryan

AU - Wu, Yi

AU - Zhou, Yuan

PY - 2011/8/29

Y1 - 2011/8/29

N2 - In 1997, Håstad showed NP-hardness of (1 - ε,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 - ε,δ)-approximating Max-3Lin(ℤ). In 2004, Khot-Kindler-Mossel- O'Donnell showed UG-hardness of (1 - ε,δ)-approximating Max-2Lin(ℤq) 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(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 - ε)-good solution over ℤ, it is hard for an algorithm to find a δ-good solution over ℤ, ℤ, or ℤm for any m ≥ q(ε, δ) of the algorithm's choosing.

AB - In 1997, Håstad showed NP-hardness of (1 - ε,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 - ε,δ)-approximating Max-3Lin(ℤ). In 2004, Khot-Kindler-Mossel- O'Donnell showed UG-hardness of (1 - ε,δ)-approximating Max-2Lin(ℤq) 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(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 - ε)-good solution over ℤ, it is hard for an algorithm to find a δ-good solution over ℤ, ℤ, or ℤm for any m ≥ q(ε, δ) of the algorithm's choosing.

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U2 - 10.1109/CCC.2011.37

DO - 10.1109/CCC.2011.37

M3 - Conference contribution

AN - SCOPUS:80051957065

SN - 9780769544113

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 23

EP - 33

BT - Proceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

T2 - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

Y2 - 8 June 2011 through 10 June 2011

ER -