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1. On Super Strong ETH

   https://doi.org/10.1007/978-3-030-24258-9_28  

   Vyas, N.; Williams, R. (January 2019, Theory and Applications of Satisfiability Testing – SAT 2019)

   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”, modeled after the ETH and the Strong ETH. We give two results on the Super-Strong ETH: 1. 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 2^kln2−Θ(1). 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 2^n(1−Ω(logk)/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, Pudlák and Zane [17]. 2. 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 2^n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2^n(1−f(k)(1−ε)/k) time for every ε>0. 

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2. Faster Random k-CNF Satisfiability

   Lincoln, Andrea; Yedidia, Adam (January 2020, Leibniz international proceedings in informatics)

   We describe an algorithm to solve the problem of Boolean CNF-Satisfiability when the input formula is chosen randomly. We build upon the algorithms of Schöning 1999 and Dantsin et al. in 2002. The Schöning algorithm works by trying many possible random assignments, and for each one searching systematically in the neighborhood of that assignment for a satisfying solution. Previous algorithms for this problem run in time O(2^(n (1- Ω(1)/k))). Our improvement is simple: we count how many clauses are satisfied by each randomly sampled assignment, and only search in the neighborhoods of assignments with abnormally many satisfied clauses. We show that assignments like these are significantly more likely to be near a satisfying assignment. This improvement saves a factor of 2^(n Ω(lg² k)/k), resulting in an overall runtime of O(2^(n (1- Ω(lg² k)/k))) for random k-SAT. 

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3. 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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4. Sometimes Hoarding is Harder than Cleaning: NP-hardness of Maximum Blocked-Clause Addition

   https://doi.org/10.29007/p5mc  

   Subercaseaux, Bernardo (May 2024, EasyChair)

   Adding blocked clauses to a CNF formula can substantially speed up SAT-solving, both in theory and practice. In theory, the addition of blocked clauses can exponentially reduce the length of the shortest refutation for a formula [17, 19]. In practice, it has been recently shown that the runtime of CDCL solvers decreases significantly for certain instance families when blocked clauses are added as a preprocessing step [10,22]. This fact is in contrast to, but not in contradiction with, prior results showing that Blocked- Clause Elimination (BCE) is sometimes an effective preprocessing step [14,15]. We suggest that the practical role of blocked clauses in SAT-solving might be richer than expected. Concretely, we propose a theoretical study of the complexity of Blocked-Clause Addition (BCA) as a preprocessing step for SAT-solving, and in particular, consider the problem of adding the maximum number of blocked clauses of a given arity k to an input formula F. While BCE is a confluent process, meaning that the order in which blocked clauses are eliminated is irrelevant, this is not the case for BCA: adding a blocked clause to a formula might unblock a different clause that was previously blocked. This order-sensitivity turns out to be a crucial obstacle for carrying out BCA efficiently as a preprocessing step. Our main result is that computing the maximum number of k-ary blocked clauses that can be added to an input formula F is NP-hard for every k ≥ 2. 

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5. 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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