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# Efficient Algorithms for Semirandom Planted CSPs at the Refutation Threshold

## February 16 @ 11:00 am - 12:00 pm

Pravesh Kothari, Princeton University

E18-304

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**Abstract**: We present an efficient algorithm to solve semi-random planted instances of any Boolean constraint satisfaction problem (CSP). The semi-random model is a hybrid between worst-case and average-case input models, where the input is generated by (1) choosing an arbitrary planted assignment x∗, (2) choosing an arbitrary clause structure, and (3) choosing literal negations for each clause from an arbitrary distribution “shifted by x∗” so that x∗ satisfies each constraint. For an n variable semi-random planted instance of a k-arity CSP, our algorithm runs in polynomial time and outputs an assignment that satisfies all but an o(1)-a fraction of constraints, provided that the instance has at least Õ (nk/2) constraints. This matches, up to polylog(n) factors, the clause threshold for algorithms that solve fully random planted CSPs [FPV15], as well as algorithms that refute random and semi-random CSPs. Our result shows that despite having a worst-case clause structure, the randomness in the literal patterns makes semi-random planted CSPs significantly easier than worst-case, where analogous results require O(n^k) constraints.

Perhaps surprisingly, our algorithm follows a different conceptual framework compared to the recent resolution of semi-random CSP refutation. This turns out to be inherent and, at a technical level, can be attributed to the need for relative spectral approximation of certain random matrices – reminiscent of the classical spectral sparsification – which ensures that an SDP can certify the uniqueness of the planted assignment. In contrast, in the refutation setting, it suffices to obtain a weaker guarantee of absolute upper bounds on the spectral norm of related matrices.

**Bio**: Pravesh Kothari is an Assistant Professor of Computer Science at Princeton University and a Visiting Professor in the School of Mathematics at the Institute for Advanced Study, Princeton. Earlier, he was an Assistant Professor at Carnegie Mellon University’s CS Department, a Postdoctoral Research Instructor at Princeton CS and the School of Math at the IAS, and obtained his Ph.D. from UT Austin in 2016. Kothari’s recent work has focused on algorithms for semi-random optimization problems via sum-of-squares semidefinite programs with connections to random matrices, extremal combinatorics, and coding theory. His research has been recognized with a Simons Award for graduate students in Theoretical Computer Science, a Google Research Scholar Award, an IIT Kanpur Young Alumnus Award, an NSF CAREER Award, and an Alfred P. Sloan Research Fellowship.