LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random (dv,dc)-regular LDPC codes are well-known to achieve the Shannon capacity of the binary symmetric channel (for sufficiently large dv and dc) 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 LP decoding algorithm of Feldman et al. which is known to correct an Omega(1/dc) fraction of errors. In this work, 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 dv and dc, a linear number of rounds of the Sherali-Adams LP hierarchy cannot correct more than an O(1/dc) fraction of errors on a random (dv,dc)-regular LDPC code. 2) For any value of dv and infinitely many values of dc, a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an O(1/dc) fraction of errors on a random (dv,dc)-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. 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 \emph{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.Comment: 23 page

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the way up to $n^{\delta}$ for some universal constant$\delta$. This shows that the $n^{O(d)}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining $\mathsf{NP}$-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively

From Weak to Strong LP Gaps for All CSPs

We study the approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations. We show that for every CSP, the approximation obtained by a basic LP relaxation, is no weaker than the approximation obtained using relaxations given by Omega(log(n)/log(log(n))) levels of the Sherali-Adams hierarchy on instances of size n. It was proved by Chan et al. [FOCS 2013] (and recently strengthened by Kothari et al. [STOC 2017]) that for CSPs, any polynomial size LP extended formulation is no stronger than relaxations obtained by a super-constant levels of the Sherali-Adams hierarchy. Combining this with our result also implies that any polynomial size LP extended formulation is no stronger than simply the basic LP, which can be thought of as the base level of the Sherali-Adams hierarchy. This essentially gives a dichotomy result for approximation of CSPs by polynomial size LP extended formulations. Using our techniques, we also simplify and strengthen the result by Khot et al. [STOC 2014] on (strong) approximation resistance for LPs. They provided a necessary and sufficient condition under which Omega(loglog n) levels of the Sherali-Adams hierarchy cannot achieve an approximation better than a random assignment. We simplify their proof and strengthen the bound to Omega(log(n)/log(log(n))) levels

Polynomial integrality gaps for strong SDP relaxations of Densest k

The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest k-subgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are i

Subsampling Mathematical Relaxations and Average-case Complexity

We initiate a study of when the value of mathematical relaxations such as linear and semidefinite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let $C$ be a family of CSPs such as 3SAT, Max-Cut, etc., and let $\Pi$ be a relaxation for $C$, in the sense that for every instance $P\in C$, $\Pi(P)$ is an upper bound the maximum fraction of satisfiable constraints of $P$. Loosely speaking, we say that subsampling holds for $C$ and $\Pi$ if for every sufficiently dense instance $P \in C$ and every $\epsilon>0$, if we let $P'$ be the instance obtained by restricting $P$ to a sufficiently large constant number of variables, then $\Pi(P') \in (1\pm \epsilon)\Pi(P)$. We say that weak subsampling holds if the above guarantee is replaced with $\Pi(P')=1-\Theta(\gamma)$ whenever $\Pi(P)=1-\gamma$. We show: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the relaxation considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every CSP under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semidefinite programs. Also there are unique CSPs for which subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semidefinite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value $1-\gamma$ have a cut value at most $1-\gamma/10$.Comment: Includes several more general results that subsume the previous version of the paper

Uncapacitated Flow-based Extended Formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems