TY - GEN
T1 - Sum-of-Squares Lower Bounds for Independent Set on Ultra-Sparse Random Graphs
AU - Kothari, Pravesh K.
AU - Potechin, Aaron
AU - Xu, Jeff
N1 - Publisher Copyright:
© 2024 Owner/Author.
PY - 2024/6/10
Y1 - 2024/6/10
N2 - We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest independent set in G is of size o(n/√d D4). In particular, degree D sum-of-squares strengthening can reduce the integrality gap of the classical theta SDP relaxation by at most a O(D4) factor. This is the first lower bound for >4-degree Sum-of-Squares (SoS) relaxation for any problems on ultra sparse random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction. Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in G(n,d/n)) that are accurate to within an absolute constant factor. All prior works lose log n factors that trivialize any lower bound on o(logn)-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any ω(1)-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.
AB - We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest independent set in G is of size o(n/√d D4). In particular, degree D sum-of-squares strengthening can reduce the integrality gap of the classical theta SDP relaxation by at most a O(D4) factor. This is the first lower bound for >4-degree Sum-of-Squares (SoS) relaxation for any problems on ultra sparse random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction. Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in G(n,d/n)) that are accurate to within an absolute constant factor. All prior works lose log n factors that trivialize any lower bound on o(logn)-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any ω(1)-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.
KW - Average-case complexity
KW - Random Matrix Theory
KW - Sum-of-Squares Lower Bounds
UR - http://www.scopus.com/inward/record.url?scp=85196615388&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85196615388&partnerID=8YFLogxK
U2 - 10.1145/3618260.3649703
DO - 10.1145/3618260.3649703
M3 - Conference contribution
AN - SCOPUS:85196615388
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1923
EP - 1934
BT - STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
A2 - Mohar, Bojan
A2 - Shinkar, Igor
A2 - O�Donnell, Ryan
PB - Association for Computing Machinery
T2 - 56th Annual ACM Symposium on Theory of Computing, STOC 2024
Y2 - 24 June 2024 through 28 June 2024
ER -