Consider a system of m polynomial equations {pi(x)=bi}i≤m of degree D≥2 in ndimensional variable x∈ℝn such that each coefficient of every pi and bis are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest m  the algorithmic threshold  for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, lowrank matrix sensing and certifying pseudorandomness of Goldreich's candidate generators and generalizations.
We show that for every d∈ℕ, the (n+m)O(d)time canonical sumofsquares (SoS) relaxation refutes such a system with high probability whenever m≥O(n)⋅(nd)D−1. We prove a lower bound in the restricted lowdegree polynomial model of computation which suggests that this tradeoff between SoS degree and the number of equations is nearly tight for all d. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree4 sumofsquares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at m≳O˜(n)⋅n(1−δ)(D−1) for 2nδtime algorithms for all δ.
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Algorithms Approaching the Threshold for Semirandom Planted Clique
We design new polynomialtime algorithms for recovering planted cliques in the semirandom graph model introduced by Feige and Kilian 2001. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices (Mehta, Mckenzie, Trevisan 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for plantedclique sizes approaching n1/2  the informationtheoretic threshold in the semirandom model (Steinhardt 2017) and a conjectured computational threshold even in the easier fullyrandom model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017.
Our algorithms are based on higher constant degree sumofsquares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite ErdősRényi random graphs into algorithms for semirandom planted clique. The use of a higherconstant degree sumofsquares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size k=o(n2/3). We also provide some evidence that the informationcomputation tradeoff of our current algorithms may be inherent by proving an averagecase lower bound for unbalanced bicliques in the lowdegreepolynomials model.
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 Award ID(s):
 2211971
 NSFPAR ID:
 10435077
 Date Published:
 Journal Name:
 ACM Symposium on Theory of Computing, STOC
 Issue:
 2023
 Page Range / eLocation ID:
 1918 to 1926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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