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1. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/S00224-022-10106-8  

   Vyas, Nikhil; Williams, R Ryan (February 2023, Theory of Computing Systems)

   Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$ $\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$ and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$$\mathcal { C}$$ $C$ , by showing that$$\mathcal { C}$$ $C$ admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class$${\mathcal C}$$ $C$ . Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of$${\sum }_{i} x_{i}$$ ${\sum }_i{x}_i$ . We show that for every sparsef, and for all “typical”$$\mathcal { C}$$ $C$ , faster#SAT algorithms for$$\mathcal { C}$$ $C$ circuits imply lower bounds against the circuit class$$f \circ \mathcal { C}$$ $f\circ C$ , which may bestrongerthan$$\mathcal { C}$$ $C$ itself. In particular:#SAT algorithms forn^k-size$$\mathcal { C}$$ $C$ -circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$ -circuits of polynomial size.#SAT algorithms for$$2^{n^{{\varepsilon }}}$$ $2^{n^{\epsilon }}$ -size$$\mathcal { C}$$ $C$ -circuits running in$$2^{n-n^{{\varepsilon }}}$$ $2^{n-n^{\epsilon }}$ time (for someε> 0) implyQuasi-NPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$ -circuits of polynomial size. Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 

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2. An Efficient ZK Compiler from SIMD Circuits to General Circuits

   https://doi.org/10.1007/s00145-024-09531-4  

   Bui, Dung; Chu, Haotian; Couteau, Geoffroy; Wang, Xiao; Weng, Chenkai; Yang, Kang; Yu, Yu (January 2025, Journal of Cryptology)

   Abstract We propose a generic compiler that can convert any zero-knowledge (ZK) proof for SIMD circuits to general circuits efficiently, and an extension that can preserve the space complexity of the proof systems. Our compiler can immediately produce new results improving upon state of the art.By plugging in our compiler to Antman, an interactive sublinear-communication protocol, we improve the overall communication complexity for general circuits from$$\mathcal {O}(C^{3/4})$$ $O\left(C^{3/4}\right)$ to$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$ . Our implementation shows that for a circuit of size$$2^{27}$$ $2^{27}$ , it achieves up to$$83.6\times $$ $83.6\times$ improvement on communication compared to the state-of-the-art implementation. Its end-to-end running time is at least$$70\%$$ $70\%$ faster in a 10Mbps network.Using the recent results on compressed$$\varSigma $$ $\Sigma$ -protocol theory, we obtain a discrete-log-based constant-round zero-knowledge argument with$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$ communication and common random string length, improving over the state of the art that has linear-size common random string and requires heavier computation.We improve the communication of a designatedn-verifier zero-knowledge proof from$$\mathcal {O}(nC/B+n^2B^2)$$ $O(nC/B+n^2{B}^2)$ to$$\mathcal {O}(nC/B+n^2)$$ $O(nC/B+n^2)$ .To demonstrate the scalability of our compilers, we were able to extract a commit-and-prove SIMD ZK from Ligero and cast it in our framework. We also give one instantiation derived from LegoSNARK, demonstrating that the idea of CP-SNARK also fits in our methodology. 

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3. Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

   https://doi.org/10.1007/s00453-023-01102-6  

   Akmal, Shyan; Chen, Lijie; Jin, Ce; Raj, Malvika; Williams, Ryan (August 2023, Algorithmica)

   Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $\tilde{O}\left(n\right)$ . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ time).Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\tilde{O}\left(n^{\lceil k/2\rceil }\right)$ (where$$k\ge 3$$ $k\ge 3$ ). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ $\tilde{O}\left(m\right)$ . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $\tilde{O}\left(n^2\right)$ . Note this is optimal, as the matrix can have$$\Omega (n^2)$$ $\Omega \left(n^2\right)$ nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$ $\tilde{O}\left(n^{2.94}\right)$ nondeterministic time algorithm.Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $2^{n/2-n/O\left(k\right)}$ . We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$ $2^{n/2}·\text{poly}\left(n\right)$ -time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$ $\#$ SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ $2^{n/2}/n^{\omega \left(1\right)}$ time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$ $2^{4n/5}·\text{poly}\left(n\right)$ . Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$ $2^{2n/3}·\text{poly}\left(n\right)$ time.Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $2^{n/3}·\text{poly}\left(n\right)$ , improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$ $2^{0.49991n}·\text{poly}\left(n\right)$ time. 

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4. Sunflowers in Set Systems with Small VC-Dimension

   https://doi.org/10.1007/s00493-025-00186-8  

   Balogh, József; Bernshteyn, Anton; Delcourt, Michelle; Ferber, Asaf; Pham, Huy Tuan (December 2025, Combinatorica)

   Abstract A family ofrdistinct sets$$\{A_1,\ldots , A_r\}$$ $\{A_1,\dots ,A_r\}$ is anr-sunflower if for all$$1 \leqslant i < j\leqslant r$$ $1⩽i<j⩽r$ and$$1 \leqslant i' < j'\leqslant r$$ $1⩽i^{\prime }<j^{\prime }⩽r$ , we have$$A_i\cap A_j = A_{i'}\cap A_{j'}$$ $A_i\cap A_j=A_{i^{\prime }}\cap A_{j^{\prime }}$ . Erdős and Rado conjectured in 1960 that every family$$\mathcal {H}$$ $H$ of$$\ell $$ $\ell$ -element sets of size at least$$K(r)^\ell $$ $K{\left(r\right)}^{\ell }$ contains anr-sunflower, whereK(r) is some function that depends only onr. We prove that if$$\mathcal {H}$$ $H$ is a family of$$\ell $$ $\ell$ -element sets of VC-dimension at mostdand$$|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell $$ $\left|H\right|>{\left(Cr(logd+{log}^{\ast }\ell )\right)}^{\ell }$ for some absolute constant$$C > 0$$ $C>0$ , then$$\mathcal {H}$$ $H$ contains anr-sunflower. This improves a recent result of Fox, Pach, and Suk. When$$d=1$$ $d=1$ , we obtain a sharp bound, namely that$$|\mathcal H| > (r-1)^\ell $$ ${\left|H\right|>(r-1)}^{\ell }$ is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest. 

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5. CP violation in K → μ+μ− with and without time dependence through a tagged analysis

   https://doi.org/10.1007/JHEP09(2025)190  

   D’Ambrosio, Giancarlo; Dery, Avital; Grossman, Yuval; Kitahara, Teppei; Marchevski, Radoslav; Santos, Diego Martínez; Schacht, Stefan (September 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We point out that using current knowledge of$$ \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) $$ $B\left(K_L^0\to {\mu }^{+}{\mu }^{-}\right)$ and$$ \mathcal{B}\left({K}_L^0\to \gamma \gamma \right) $$ $B\left(K_L^0\to \mathrm{\gamma \gamma }\right)$ , one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry,A_CP(K⁰→ μ⁺μ⁻), and of$$ \mathcal{B}\left({K}_S^0\to {\mu}^{+}{\mu}^{-}\right) $$ $B\left(K_S^0\to {\mu }^{+}{\mu }^{-}\right)$ . We discuss the interplay between this set of observables, and demonstrate that determining sign[A_CP(K⁰→μ⁺μ⁻)] would eliminate the discrete ambiguity in the Standard Model prediction for$$ \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) $$ $B\left(K_L^0\to {\mu }^{+}{\mu }^{-}\right)$ . We then move on to feasibility studies within an LHCb-like setup, using both time-integrated and time-dependent information, employingK⁰and$$ {\overline{K}}^0 $$ ${\overline{K}}^0$ tagging methods. We find that, within an optimistic scenario, the short-distance amplitude, proportional to the CKM parameter combination$$ \mid {A}^2{\lambda}^5\overline{\eta}\mid $$ $\mid A^2{\lambda }^5\overline{\eta }\mid$ , could be constrained by LHCb at the level of about 35% of its Standard Model value, and the discrete ambiguity in$$ \mathcal{B}{\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right)}_{\textrm{SM}} $$ $B{\left(K_L^0\to {\mu }^{+}{\mu }^{-}\right)}_{\mathrm{SM}}$ could be resolved at more than 3σby the end of the high luminosity LHC. 

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