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1. Local Correction of Linear Functions over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (June 2024, Association for Computing Machinery)

   We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls. 

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2. Low Degree Local Correction Over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (November 2024, ACM Corr)

   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 O˜d((logn)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 the authors [ABPSS24] (STOC 2024) who considered the case of linear polynomials 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 [ABPSS24] from linear 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 [ABPSS24] 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. 

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3. Low Degree Local Correction Over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (January 2025, Society for Industrial and Applied Mathematics)

   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. 

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4. Low Degree Local Correction Over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (January 2025, Society for Industrial and Applied Mathematics)

   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. 

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5. New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

   Compton, Spencer; Mitrovic, Slobodan; Rubinfeld, Ronitt (July 2024, Algorithmica)

   Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For (1+ε)-approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with O(lognε) update and O(logn) query worst-case time. Further, we design a local computation algorithm that uses only O(logNε) queries when all jobs are length at least 1 and have starting/ending times within [0,N]. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are poly(logn,1ε). Equivalently, this is the first algorithm that maintains a (1+ε)-approximation of the maximum independent set of a collection of weighted intervals in poly(logn,1ε) time updates/queries. This is an exponential improvement in 1/ε over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese ~[SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness. We also extend our approaches for interval scheduling on a single machine to examine the setting with M machines. 

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