Pravesh Kothari

# Strongly Refuting all Semirandom Boolean CSPs

We give an efficient algorithm to strongly refute semi-random instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best-known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean $$k$$-XOR problem on $$n$$ variables that have $$\tilde O(n^{k/2})$$ constraints. (In a semi-random $$k$$-XOR instance, the equations can be arbitrary and only the right-hand sides are random.)

One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work. Our approach involves taking an instance that does not satisfy this property (i.e., is not pseudorandom) and reducing it to a partitioned collection of 2-XOR instances. We analyze these subinstances using a carefully chosen quadratic form as a proxy, which in turn is bounded via a combination of spectral methods and semidefinite programming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernstein inequality. Even for the purely random case, this leads to a shorter proof compared to the ones in the literature that rely on problem-specific trace-moment computations.