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1. 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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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. Solving Random Planted CSPs Below the n^{k/2} Threshold

   https://doi.org/10.4230/lipics.icalp.2026.23  

   Basu, Arpon; Hsieh, Jun-Ting; Lin, Andrew D; Manohar, Peter (January 2026, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bhattacharya, Sayan; Nanongkai, Danupon; Benedikt, Michael; Puppis, Gabriele (Ed.)

   Abstract: We present a family of algorithms to solve random planted instances of any k-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment x^*, and then (2) sampling constraints from a particular "planting distribution" designed so that x^* will satisfy every constraint. Given an n variable instance of a k-ary Boolean CSP with m constraints, our algorithm runs in time n^O(𝓁) for a choice of a parameter 𝓁, and succeeds in outputting a satisfying assignment if m ⩾ O(n)⋅(n/𝓁)^{k/2 - 1} log n. This generalizes the poly(n)-time algorithm of [Vitaly Feldman et al., 2015], the case of 𝓁 = O(1), to larger runtimes, and matches the constraint number vs. runtime trade-off established for refuting random CSPs by [Prasad Raghavendra et al., 2017]. Our algorithm is conceptually different from the recent algorithm of [Venkatesan Guruswami et al., 2023], which gave a poly(n)-time algorithm to solve semirandom CSPs with m ⩾ Õ(n^{k/2}) constraints by exploiting conditions that allow a basic SDP to recover the planted assignment x^* exactly. Instead, we forego certificates of uniqueness and recover x^* in two steps: we first use a degree-O(𝓁) Sum-of-Squares SDP to find some x̂ that is o(1)-close to x^*, and then we use a second rounding procedure to recover x^* from x̂. 

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4. Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering

   https://doi.org/10.4230/lipics.esa.2025.62  

   Bagheri_Nezhad, Sina; Bandyapadhyay, Sayan; Chen, Tianzhi (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Benoit, Anne; Kaplan, Haim; Wild, Sebastian; Herman, Grzegorz (Ed.)

   In a seminal work, Chierichetti et al. [Chierichetti et al., 2017] introduced the (t,k)-fair clustering problem: Given a set of red points and a set of blue points in a metric space, a clustering is called fair if the number of red points in each cluster is at most t times and at least 1/t times the number of blue points in that cluster. The goal is to compute a fair clustering with at most k clusters that optimizes certain objective function. Considering this problem, they designed a polynomial-time O(1)- and O(t)-approximation for the k-center and the k-median objective, respectively. Recently, Carta et al. [Carta et al., 2024] studied this problem with the sum-of-radii objective and obtained a (6+ε)-approximation with running time O((k log_{1+ε}(k/ε))^k n^O(1)), i.e., fixed-parameter tractable in k. Here n is the input size. In this work, we design the first polynomial-time O(1)-approximation for (t,k)-fair clustering with the sum-of-radii objective, improving the result of Carta et al. Our result places sum-of-radii in the same group of objectives as k-center, that admit polynomial-time O(1)-approximations. This result also implies a polynomial-time O(1)-approximation for the Euclidean version of the problem, for which an f(k)⋅n^O(1)-time (1+ε)-approximation was known due to Drexler et al. [Drexler et al., 2023]. Here f is an exponential function of k. We are also able to extend our result to any arbitrary 𝓁 ≥ 2 number of colors when t = 1. This matches known results for the k-center and k-median objectives in this case. The significant disparity of sum-of-radii compared to k-center and k-median presents several complex challenges, all of which we successfully overcome in our work. Our main contribution is a novel cluster-merging-based analysis technique for sum-of-radii that helps us achieve the constant-approximation bounds. 

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5. Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints

   https://doi.org/10.4230/lipics.approx/random.2025.23  

   Bandyapadhyay, Sayan; Chen, Tianzhi (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ene, Alina; Chattopadhyay, Eshan (Ed.)

   In this work, we study k-min-sum-of-radii (k-MSR) clustering under mergeable constraints. k-MSR seeks to group data points using a set of up to k balls, such that the sum of the radii of the balls is minimized. A clustering constraint is called mergeable if merging two clusters satisfying the constraint, results in a cluster that also satisfies the constraint. Many popularly studied constraints are mergeable, including fairness constraints and lower bound constraints. In our work, we design a (4+ε)-approximation for k-MSR under any given mergeable constraint with runtime 2^{O(k/(ε)⋅log²k/ε)} n⁴, i.e., fixed-parameter tractable in k for constant ε. Our result directly improves upon the FPT (6+ε)-approximation by Carta et al. [Carta et al., 2024]. We also provide a hardness result that excludes the exact solvability of k-MSR under any given mergeable constraint in time f(k)n^o(k), assuming ETH is true. 

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