arXiv subjects

Aaron L. Putterman

Publications and source records attributed to Aaron L. Putterman.

4 recordsLinked to original sources

arXiv2024

Bivariate Linear Operator Codes

In this work, we present a generalization of the linear operator family of codes that captures more codes that achieve list decoding capacity. Linear operator (LO) codes were introduced by Bhandari, Harsha, Kumar, and Sudan [BHKS24] as a way to capture capacity-achieving codes. In their framework, a code is specified by a collection of linear operators that are applied to a message polynomial and then evaluated at a specified set of evaluation points. We generalize this idea in a way that can be applied to bivariate message polynomials, getting what we call bivariate linear operator (B-LO) codes. We show that bivariate linear operator codes capture more capacity-achieving codes, including permuted product codes introduced by Berman, Shany, and Tamo [BST24]. These codes work with bivariate message polynomials, which is why our generalization is necessary to capture them as a part of the linear operator framework. Similarly to the initial paper on linear operator codes, we present sufficient conditions for a bivariate linear operator code to be list decodable. Using this characterization, we are able to derive the theorem characterizing list-decodability of LO codes as a specific case of our theorem for B-LO codes. We also apply this theorem to show that permuted product codes are list decodable up to capacity, thereby unifying this result with those of known list-decodable LO codes, including Folded Reed-Solomon, Multiplicity, and Affine Folded Reed-Solomon codes.

cs.IT↗

arXiv2024

Near-optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

A $(1 \pm \epsilon)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \pm \epsilon)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \pm \epsilon)$-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \emph{linear sketch}) over the hyperedges of $G$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $\widetilde{O}(n r \log(m) / \epsilon^2)$ bits which with high probability contains sufficient information to recover a $(1 \pm \epsilon)$ cut-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges for any hypergraph with at most $m$ edges each of which has arity bounded by $r$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $\widetilde{O}(n r^2 \log^4(m) / \epsilon^2)$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $\epsilon \in (0,1)$, one needs $\Omega(nr \log(m/n) / \log(n))$ bits to construct a $(1 \pm \epsilon)$-sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $r$ and $\log(m)$.

cs.DS↗

arXiv2024

Efficient Algorithms and New Characterizations for CSP Sparsification

CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: how much can an instance of a constraint satisfaction problem be sparsified (by retaining a reweighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification captures as a special case several well-studied problems including graph cut-sparsification, hypergraph cut-sparsification, hypergraph XOR-sparsification, and corresponds to a general class of hypergraph sparsification problems where an arbitrary $0/1$-valued {\em splitting function} is used to define the notion of cutting a hyperedge (see, for instance, Veldt-Benson-Kleinberg SIAM Review 2022). The main question here is to understand, for a given constraint predicate $P:\Sigma^r \to \{0,1\}$ (where variables are assigned values in $\Sigma$), the smallest constant $c$ such that $\widetilde{O}(n^c)$ sized sparsifiers exist for every instance of a constraint satisfaction problem over $P$. A recent work of Khanna, Putterman and Sudan (SODA 2024) [KPS24] showed {\em existence} of near-linear size sparsifiers for new classes of CSPs. In this work (1) we significantly extend the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications; (2) we give a polynomial-time algorithm to extract such sparsifications for all the problems we study including the first efficient sparsification algorithms for the problems studied in [KPS24].

cs.DS↗

arXiv2024

Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs

Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.

cs.DS↗