The Biased Homogeneous r-Lin Problem

Author Suprovat Ghoshal

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Suprovat Ghoshal
  • University of Michigan, Ann Arbor, MI, USA


Most of this work was carried out when the author was visiting KTH, Stockholm. The author thanks Johan Håstad for inviting him to visit KTH, for the many insightful discussions and ideas that have played a key role in this work, and for his comments on a previous draft of the manuscript. The author thanks Per Austrin for several useful discussions on related topics. The author also thanks the anonymous reviewers for several helpful suggestions on the manuscript, and for fixing an issue in a previous version of the work.

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Suprovat Ghoshal. The Biased Homogeneous r-Lin Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The p-biased Homogeneous r-Lin problem (Hom-r-Lin_p) is the following: given a homogeneous system of r-variable equations over m{F}₂, the goal is to find an assignment of relative weight p that satisfies the maximum number of equations. In a celebrated work, Håstad (JACM 2001) showed that the unconstrained variant of this i.e., Max-3-Lin, is hard to approximate beyond a factor of 1/2. This is also tight due to the naive random guessing algorithm which sets every variable uniformly from {0,1}. Subsequently, Holmerin and Khot (STOC 2004) showed that the same holds for the balanced Hom-r-Lin problem as well. In this work, we explore the approximability of the Hom-r-Lin_p problem beyond the balanced setting (i.e., p ≠ 1/2), and investigate whether the (p-biased) random guessing algorithm is optimal for every p. Our results include the following: - The Hom-r-Lin_p problem has no efficient 1/2 + 1/2 (1 - 2p)^{r-2} + ε-approximation algorithm for every p if r is even, and for p ∈ (0,1/2] if r is odd, unless NP ⊂ ∪_{ε>0}DTIME(2^{n^ε}). - For any r and any p, there exists an efficient 1/2 (1 - e^{-2})-approximation algorithm for Hom-r-Lin_p. We show that this is also tight for odd values of r (up to o_r(1)-additive factors) assuming the Unique Games Conjecture. Our results imply that when r is even, then for large values of r, random guessing is near optimal for every p. On the other hand, when r is odd, our results illustrate an interesting contrast between the regimes p ∈ (0,1/2) (where random guessing is near optimal) and p → 1 (where random guessing is far from optimal). A key technical contribution of our work is a generalization of Håstad’s 3-query dictatorship test to the p-biased setting.

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Biased Approximation Resistance
  • Constraint Satisfaction Problems


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