SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√{k(n-k)}) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function.
Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf-Yuen conjecture and prove that the SoS rank for SQFs is at most O(√{nk}log(n)).
We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√nlog(n)) SoS certificate for this problem.
symmetric quadratic functions
SoS certificate
hypercube optimization
semidefinite programming
Theory of computation~Semidefinite programming
Theory of computation~Convex optimization
90:1-90:20
Track A: Algorithms, Complexity and Games
Adam
Kurpisz
Adam Kurpisz
Department of Mathematics, ETH Zürich, Switzerland
supported by SNSF project PZ00P2_174117.
Aaron
Potechin
Aaron Potechin
Department of Computer Science, University of Chicago, IL, USA
supported in part by NSF grant CCF-2008920.
Elias Samuel
Wirth
Elias Samuel Wirth
Institute of Mathematics, TU Berlin, Germany
10.4230/LIPIcs.ICALP.2021.90
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Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth
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