Revised: November 7, 2021
Published: February 14, 2022
Abstract: [Plain Text Version]
We prove almost optimal hardness for Max k-CSPR. In Max k-CSPR, we are given a set of constraints, each of which depends on at most k variables. Each variable can take any value from 1,2,…,R. The goal is to find an assignment to variables that maximizes the number of satisfied constraints.
We show that, for any k≥2 and R≥16, it is NP-hard to approximate Max k-CSPR to within factor kO(k)(logR)k/2/Rk−1. In the regime where 3≤k=o(logR/loglogR), this ratio improves upon Chan's O(k/Rk−2) factor NP-hardness of approximation of Max k-CSPR (J. ACM 2016). Moreover, when k=2, our result matches the best known hardness result of Khot, Kindler, Mossel and O'Donnell (SIAM J. Comp. 2007). We remark here that NP-hardness of an approximation factor of 2O(k)log(kR)/Rk−1 is implicit in the (independent) work of Khot and Saket (ICALP 2015), which is better than our ratio for all k≥3.
In addition to the above hardness result, by extending an algorithm for Max 2-CSPR by Kindler, Kolla and Trevisan (SODA 2016), we provide an Ω(logR/Rk−1)-approximation algorithm for Max k-CSPR. Thanks to Khot and Saket's result, this algorithm is tight up to a factor of O(k2) when k≤RO(1). In comparison, when 3≤k is a constant, the previously best known algorithm achieves an O(k/Rk−1)-approximation for the problem, which is a factor of O(klogR) from the inapproximability ratio in contrast to our gap of O(k2).
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A conference version of this paper appeared in the Proceedings of the 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'16).