LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Abstract

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , 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 $\Omega (1/d_{c})$ 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 $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing 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 balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness 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 $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.