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1. On Super Strong ETH

   https://doi.org/10.1613/jair.1.11859  

   Vyas, Nikhil; Williams, Ryan (January 2021, Journal of Artificial Intelligence Research)

   Multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield a 2n(1-1/O(k)) time algorithm for k-SAT in the worst case. For this reason, it has been hypothesized that the worst-case k-SAT problem cannot be solved in 2n(1-f(k)/k) time for any unbounded function f. This hypothesis has been called the "Super-Strong ETH", modelled after the ETH and the Strong ETH. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the "critical threshold", where the clause-to-variable ratio is such that randomly chosen instances are satisfiable with probability 1/2. We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. For example, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in  2n(1-c*log(k)/k) time with high probability (over the choice of the formula and the randomness of the algorithm). It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1999).   The Unique k-SAT problem is the special case where there is at most one satisfying assignment. Improving prior reductions, we show that the Super-Strong ETHs for Unique k-SAT and k-SAT are equivalent. More precisely, we show the time complexities of Unique k-SAT and k-SAT are very tightly correlated: if Unique k-SAT is in  2n(1-f(k)/k) time for an unbounded f, then k-SAT is in 2n(1-f(k)/(2k)) time. 

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2. Conditional lower bounds for sparse parameterized 2-CSP: A streamlined proof

   Karthik, C_S; Marx, Dániel; Pilipczuk, Marcin; Souza, Uéverton (January 2024, Society for Industrial and Applied Mathematics)

   Assuming the Exponential Time Hypothesis (ETH), a result of Marx (ToC’10) implies that there is no f (k) · n^o(k/ log k) time algorithm that can solve 2-CSPs with k constraints (over a domain of arbitrary large size n) for any computable function f . This lower bound is widely used to show that certain parameterized problems cannot be solved in time f (k) · n^o(k/ log k) time (assuming the ETH). The purpose of this note is to give a streamlined proof of this result. 

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3. Matrix Multiplication Verification Using Coding Theory

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.42  

   Bennett, Huck; Gajulapalli, Karthik; Golovnev, Alexander; Warton, Evelyn (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   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). 

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4. Local Enumeration and Majority Lower Bounds

   https://doi.org/10.4230/LIPIcs.CCC.2024.17  

   Gurumukhani, Mohit; Paturi, Ramamohan; Pudlák, Pavel; Saks, Michael; Talebanfard, Navid (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics. 

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5. On Sketching Approximations for Symmetric Boolean CSPs

   Boyland, Joanna; Hwang, Michael; Prasad, Tarun; Singer, Noah; Velusamy, Santhoshini (September 2022, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Chakrabarti, Amit; Swamy, Chaitanya (Ed.)

   A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a predicate f:{-1,1}^k → {0,1}. An n-variable instance of Max-CSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ n-space streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ n-space sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, Max-CSP(f) is (α(f)-ε)-approximable by an O(log n)-space linear sketching algorithm, but (α(f)+ε)-approximation sketching algorithms require Ω(√n) space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2-o(1))2^{-k}-approximated by O(log n)-space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{-k}-approximations require Ω(n) space! We also resolve the ratio for the "at-least-(k-1)-1’s" function for all even k; the "exactly-(k+1)/2-1’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closed-form expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddle-point" properties of this dichotomy. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ n-space streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})-ε)-approximations in o(√ n) space. 

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