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We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of (Buchbinder, Feldman, Seffi, and Schwartz [14] and Censor-Hillel, Levy, and Shachnai [16]), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In n-vertex graphs, for adversarially ordered streams, our algorithm uses O (n1-Ω(1)) (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than 1/2. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than 1/2 is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun [37]) and space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan [35]). In terms of techniques, our algorithms partition the vertices into a small number of different types based on the structure of their local neighborhood, ensuring that each type carries enough information about the structure to approximately simulate the local algorithm on a vertex with that type. We then develop tools to accurately estimate the frequency of each type. This allows us to simulate an execution of the local algorithm on all vertices, and thereby approximate the value of the maximum directed cut. 

We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of (Buchbinder, Feldman, Seffi, and Schwartz [14] and Censor-Hillel, Levy, and Shachnai [16]), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In n-vertex graphs, for adversarially ordered streams, our algorithm uses O (n1-Ω(1)) (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than 1/2. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than 1/2 is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun [37]) and space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan [35]). In terms of techniques, our algorithms partition the vertices into a small number of different types based on the structure of their local neighborhood, ensuring that each type carries enough information about the structure to approximately simulate the local algorithm on a vertex with that type. We then develop tools to accurately estimate the frequency of each type. This allows us to simulate an execution of the local algorithm on all vertices, and thereby approximate the value of the maximum directed cut. 

In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results. Low-degree polynomials over the Boolean cube {0,1}n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [1] from linear polynomials to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [1] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results. Low-degree polynomials over the Boolean cube {0,1}n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [1] from linear polynomials to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [1] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

In this paper, we construct new t-server Private Information Retrieval (PIR) schemes with communication complexity subpolynomial in the previously best known, for all but finitely many t. Our results are based on combining derivatives (in the spirit of Woodruff-Yekhanin) with the Matching Vector based PIRs of Yekhanin and Efremenko. Previously such a combination was achieved in an ingenious way by Dvir and Gopi, using polynomials and derivatives over certain exotic rings, en route to their fundamental result giving the first 2-server PIR with subpolynomial communication. Our improved PIRs are based on two ingredients: - We develop a new and direct approach to combine derivatives with Matching Vector based PIRs. This approach is much simpler than that of Dvir-Gopi: it works over the same field as the original PIRs, and only uses elementary properties of polynomials and derivatives. - A key subproblem that arises in the above approach is a higher-order polynomial interpolation problem. We show how "sparse S-decoding polynomials", a powerful tool from the original constructions of Matching Vector PIRs, can be used to solve this higher-order polynomial interpolation problem using surprisingly few higer-order evaluations. Using the known sparse S-decoding polynomials, in combination with our ideas leads to our improved PIRs. Notably, we get a 3-server PIR scheme with communication 2O∼((logn)1/3), improving upon the previously best known communication of 2O∼(logn√) due to Efremenko. 

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 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 $$\tilde{O}(nr\log(m)/\epsilon^{2})$$ 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 $$\tilde{O}(nr^{2}\log^{4}({m})/\epsilon^{2})$$ 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 $$\log(m)$$. The starting point for our improved algorithm is importance sampling of hyperedges based on the new notion of $$k$$ -cut strength introduced in the recent work of Quanrud (SODA 2024). The natural algorithm based on this concept leads to $$\log m$$ levels of sampling where errors can potentially accumulate, and this accounts for the polylog $(m)$ losses in the sketch size of the natural algorithm. We develop a more intricate analysis of the accumulation in error to show most levels do not contribute to the error and actual loss is only polylog $(n)$. Combining with careful preprocessing (and analysis) this enables us to get rid of all extraneous $$\log m$$ factors in the sketch size, but the quadratic dependence on $$r$$ remains. This dependence originates from use of correlated $$\ell_{0}$$ -samplers to recover a large number of low-strength edges in a hypergraph simultaneously by looking at neighborhoods of individual vertices. In graphs, this leads to discovery of $$\Omega(n)$$ edges in a single shot, whereas in hypergraphs, this may potentially only reveal $$O$$($$n$$/$$r$$) new edges, thus requiring $$\Omega(r)$$ rounds of recovery. To remedy this we introduce a new technique of random fingerprinting of hyperedges which effectively eliminates the correlations created by large arity hyperedges, and leads to a scheme for recovering hyperedges of low strength with an optimal dependence on $$r$$. Putting all these ingredients together yields our linear sketching algorithm. Our lower bound is established by a reduction from the universal relation problem in the one-way communication setting. 

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 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 $$\tilde{O}(nr\log(m)/\epsilon^{2})$$ 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 $$\tilde{O}(nr^{2}\log^{4}({m})/\epsilon^{2})$$ 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 $$\log(m)$$. The starting point for our improved algorithm is importance sampling of hyperedges based on the new notion of $$k$$ -cut strength introduced in the recent work of Quanrud (SODA 2024). The natural algorithm based on this concept leads to $$\log m$$ levels of sampling where errors can potentially accumulate, and this accounts for the polylog $(m)$ losses in the sketch size of the natural algorithm. We develop a more intricate analysis of the accumulation in error to show most levels do not contribute to the error and actual loss is only polylog $(n)$. Combining with careful preprocessing (and analysis) this enables us to get rid of all extraneous $$\log m$$ factors in the sketch size, but the quadratic dependence on $$r$$ remains. This dependence originates from use of correlated $$\ell_{0}$$ -samplers to recover a large number of low-strength edges in a hypergraph simultaneously by looking at neighborhoods of individual vertices. In graphs, this leads to discovery of $$\Omega(n)$$ edges in a single shot, whereas in hypergraphs, this may potentially only reveal $$O$$($$n$$/$$r$$) new edges, thus requiring $$\Omega(r)$$ rounds of recovery. To remedy this we introduce a new technique of random fingerprinting of hyperedges which effectively eliminates the correlations created by large arity hyperedges, and leads to a scheme for recovering hyperedges of low strength with an optimal dependence on $$r$$. Putting all these ingredients together yields our linear sketching algorithm. Our lower bound is established by a reduction from the universal relation problem in the one-way communication setting.