LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random <inline-formula> <tex-math notation="LaTeX">$(d_{v},d_{c})$ </tex-math></inline-formula>-<italic>regular</italic> low-density parity-check (LDPC) codes, where each variable is involved in <inline-formula> <tex-math notation="LaTeX">$d_{v}$ </tex-math></inline-formula> parity checks and each parity check involves <inline-formula> <tex-math notation="LaTeX">$d_{c}$ </tex-math></inline-formula> variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large <inline-formula> <tex-math notation="LaTeX">$d_{v}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d_{c}$ </tex-math></inline-formula>, under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an <inline-formula> <tex-math notation="LaTeX">$\Omega (1/d_{c})$ </tex-math></inline-formula> fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of <inline-formula> <tex-math notation="LaTeX">$d_{v}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d_{c}$ </tex-math></inline-formula>, a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an <inline-formula> <tex-math notation="LaTeX">$O(1/d_{c})$ </tex-math></inline-formula> fraction of errors on a random <inline-formula> <tex-math notation="LaTeX">$(d_{v},d_{c})$ </tex-math></inline-formula>-regular LDPC code; and 2) for any value of <inline-formula> <tex-math notation="LaTeX">$d_{v}$ </tex-math></inline-formula> and infinitely many values of <inline-formula> <tex-math notation="LaTeX">$d_{c}$ </tex-math></inline-formula>, a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an <inline-formula> <tex-math notation="LaTeX">$O(1/d_{c})$ </tex-math></inline-formula> fraction of errors on a random <inline-formula> <tex-math notation="LaTeX">$(d_{v},d_{c})$ </tex-math></inline-formula>-regular LDPC code. Our proofs use a new <italic>stretching</italic> and <italic>collapsing</italic> technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special <italic>balanced pairwise independent distributions</italic> for Sherali–Adams and special <italic>cosets of balanced pairwise independent subgroups</italic> for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{q}^{d}$ </tex-math></inline-formula> (for <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> any power of 2 and <inline-formula> <tex-math notation="LaTeX">$d = 2,3$ </tex-math></inline-formula>) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that <italic>expansion</italic> seems to be both the <italic>strength</italic> and the <italic>weakness</italic> of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Hypergraph Vertex Cover, we obtain an improved integrality gap of <inline-formula> <tex-math notation="LaTeX">$k-1-\epsilon $ </tex-math></inline-formula> that holds after a <italic>linear</italic> number of rounds of the Lasserre hierarchy, for any <inline-formula> <tex-math notation="LaTeX">$k = q+1$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> an arbitrary prime power. The best previous gap for a linear number of rounds was equal to <inline-formula> <tex-math notation="LaTeX">$2-\epsilon $ </tex-math></inline-formula> and due to Schoenebeck.