arXiv subjects

Andrew D. Lin

Publications and source records attributed to Andrew D. Lin.

3 recordsLinked to original sources

arXiv2026

An Approximate Cauchy-Schwarz Inequality and Improved Bounds for Sherali-Adams Refutation of Semirandom CSPs

We formulate an approximate Cauchy-Schwarz inequality and show that it is satisfied by solutions to the Sherali-Adams linear programming hierarchy (interpreted as ``pseudo-distributions''). As a consequence, we resolve a question left open by the work of O'Donnell and Schramm [OS19] that they had explicitly attributed to the lack of such an inequality. A Cauchy-Schwarz inequality is exactly satisfied by pseudo-distributions satisfying the constraints of the sum-of-squares semidefinite programming hierarchy and already has scores of applications. However, the proof there requires global positive semidefiniteness. Our approximate version, on the other hand, relies only on local positive semidefiniteness satisfied by the Sherali-Adams pseudo-distributions. Our formulation loses an additive error that scales with the L1 norm of the coefficients of the constituent polynomials, and this loss is asymptotically tight. Our proof is elementary and relies on a simple sampling argument. As an application, we resolve a question left open in the work of O'Donnell and Schramm that gives a trade-off between constraint density and the Sherali-Adams degree for refuting random constraint satisfaction problems. Specifically, for odd arity CSPs, we show that the constraint density requirement for a given degree can be improved by a polynomial factor in $n$. Along the way, we observe that by a simple extension, the results in their work extend to a more general semirandom setting.

cs.CC↗

arXiv2025

Solving Random Planted CSPs below the $n^{k/2}$ Threshold

We present a family of algorithms to solve random planted instances of any $k$-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment $x^*$, and then (2) sampling constraints from a particular "planting distribution" designed so that $x^*$ will satisfy every constraint. Given an $n$ variable instance of a $k$-ary Boolean CSP with $m$ constraints, our algorithm runs in time $n^{O(\ell)}$ for a choice of a parameter $\ell$, and succeeds in outputting a satisfying assignment if $m \geq O(n) \cdot (n/\ell)^{\frac{k}{2} - 1} \log n$. This generalizes the $\mathrm{poly}(n)$-time algorithm of [FPV15], the case of $\ell = O(1)$, to larger runtimes, and matches the constraint number vs.\ runtime trade-off established for refuting random CSPs by [RRS17]. Our algorithm is conceptually different from the recent algorithm of [GHKM23], which gave a $\mathrm{poly}(n)$-time algorithm to solve semirandom CSPs with $m \geq \tilde{O}(n^{\frac{k}{2}})$ constraints by exploiting conditions that allow a basic SDP to recover the planted assignment $x^*$ exactly. Instead, we forego certificates of uniqueness and recover $x^*$ in two steps: we first use a degree-$O(\ell)$ Sum-of-Squares SDP to find some $\hat{x}$ that is $o(1)$-close to $x^*$, and then we use a second rounding procedure to recover $x^*$ from $\hat{x}$.

cs.DS↗

arXiv2024

Improved Lower Bounds for all Odd-Query Locally Decodable Codes

We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$. Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees. We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees.

cs.CC↗