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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time).
To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm).
Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T.
We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).

Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2024.42, author = {Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn}, title = {{Matrix Multiplication Verification Using Coding Theory}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.42}, URN = {urn:nbn:de:0030-drops-210352}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.42}, annote = {Keywords: Matrix Multiplication Verification, Derandomization, Sparse Matrices, Error-Correcting Codes, Hardness Barriers, Reductions} }

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**Published in:** LIPIcs, Volume 304, 5th Conference on Information-Theoretic Cryptography (ITC 2024)

We study the problem of function inversion with preprocessing where, given a function f : [N] → [N] and a point y in its image, the goal is to find an x such that f(x) = y using at most T oracle queries to f and S bits of preprocessed advice that depend on f.
The seminal work of Corrigan-Gibbs and Kogan [TCC 2019] initiated a line of research that shows many exciting connections between the non-adaptive setting of this problem and other areas of theoretical computer science. Specifically, they introduced a very weak class of algorithms (strongly non-adaptive) where the points queried by the oracle depend only on the inversion point y, and are independent of the answers to the previous queries and the S bits of advice. They showed that proving even mild lower bounds on strongly non-adaptive algorithms for function inversion would imply a breakthrough result in circuit complexity.
We prove that every strongly non-adaptive algorithm for function inversion (and even for its special case of permutation inversion) must have ST = Ω(N log (N) log (T)). This gives the first improvement to the long-standing lower bound of ST = Ω(N log N) due to Yao [STOC 90]. As a corollary, we conclude the first separation between strongly non-adaptive and adaptive algorithms for permutation inversion, where the adaptive algorithm by Hellman [TOIT 80] achieves the trade-off ST = O(N log N).
Additionally, we show equivalence between lower bounds for strongly non-adaptive data structures and the one-way communication complexity of certain partial functions. As an example, we recover our lower bound on function inversion in the communication complexity framework.

Karthik Gajulapalli, Alexander Golovnev, and Samuel King. On the Power of Adaptivity for Function Inversion. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajulapalli_et_al:LIPIcs.ITC.2024.5, author = {Gajulapalli, Karthik and Golovnev, Alexander and King, Samuel}, title = {{On the Power of Adaptivity for Function Inversion}}, booktitle = {5th Conference on Information-Theoretic Cryptography (ITC 2024)}, pages = {5:1--5:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-333-1}, ISSN = {1868-8969}, year = {2024}, volume = {304}, editor = {Aggarwal, Divesh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2024.5}, URN = {urn:nbn:de:0030-drops-205137}, doi = {10.4230/LIPIcs.ITC.2024.5}, annote = {Keywords: Function Inversion, Non-Adaptive lower bounds, Communication Complexity} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit 𝖢: {0,1}ⁿ → {0,1}^m, m > n, the task is to find a y ∈ {0,1}^m outside the range of 𝖢. For an integer k ≥ 2, NC⁰_k-Avoid is a special case of Avoid where each output bit of 𝖢 depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC⁰₄-Avoid, thus establishing conditional hardness of the NC⁰₄-Avoid problem. On the other hand, NC⁰₂-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC⁰₃-Avoid open.
We give the first reduction of an explicit construction question to NC⁰₃-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC⁰₃-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits.
We also give deterministic polynomial-time algorithms for all NC⁰_k-Avoid problems for m ≥ n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ≥ n^{k-1}.

Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi. Range Avoidance for Constant Depth Circuits: Hardness and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 65:1-65:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gajulapalli_et_al:LIPIcs.APPROX/RANDOM.2023.65, author = {Gajulapalli, Karthik and Golovnev, Alexander and Nagargoje, Satyajeet and Saraogi, Sidhant}, title = {{Range Avoidance for Constant Depth Circuits: Hardness and Algorithms}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {65:1--65:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.65}, URN = {urn:nbn:de:0030-drops-188901}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.65}, annote = {Keywords: Boolean function analysis, Explicit Constructions, Low-depth Circuits, Range Avoidance, Matrix Rigidity, Circuit Lower Bounds} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R₁ and R₂. In round R₁, the capacity of each school is fixed and mechanism M₁ finds a student optimal stable matching. In round R₂, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations.
It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results: the first simply disallows any re-allocations in round R₂, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round R₂ under certain settings.

Karthik Gajulapalli, James A. Liu, Tung Mai, and Vijay V. Vazirani. Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gajulapalli_et_al:LIPIcs.FSTTCS.2020.21, author = {Gajulapalli, Karthik and Liu, James A. and Mai, Tung and Vazirani, Vijay V.}, title = {{Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.21}, URN = {urn:nbn:de:0030-drops-132626}, doi = {10.4230/LIPIcs.FSTTCS.2020.21}, annote = {Keywords: stable matching, mechanism design, NP-Hardness} }

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