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1. 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 (February 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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2. L-Systems and the Lovász Number

   https://doi.org/10.1007/s00493-025-00136-4  

   Linz, William (April 2025, Combinatorica)

   Abstract Given integers$$n> k > 0$$ $n>k>0$, and a set of integers$$L \subset [0, k-1]$$ $L\subset [0,k-1]$, anL-systemis a family of sets$$\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) $$ $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$such that$$|F \cap F'| \in L$$ $|F\cap F^{\prime }|\in L$for distinct$$F, F'\in \mathcal {F}$$ $F,F^{\prime }\in F$.L-systems correspond to independent sets in a certain generalized Johnson graphG(n, k, L), so that the maximum size of anL-system is equivalent to finding the independence number of the graphG(n, k, L). TheLovász number$$\vartheta (G)$$ $\vartheta \left(G\right)$is a semidefinite programming approximation of the independence number$$\alpha $$ $\alpha$of a graphG. In this paper, we determine the leading order term of$$\vartheta (G(n, k, L))$$ $\vartheta \left(G\right(n,k,L\left)\right)$of any generalized Johnson graph withkandLfixed and$$n\rightarrow \infty $$ $n\to \infty$. As an application of this theorem, we give an explicit construction of a graphGonnvertices with a large gap between the Lovász number and the Shannon capacityc(G). Specifically, we prove that for any$$\epsilon > 0$$ $ϵ>0$, for infinitely manynthere is a generalized Johnson graphGonnvertices which has ratio$$\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/c\left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which improves on all known constructions. The graphGa fortiorialso has ratio$$\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/\alpha \left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which greatly improves on the best known explicit construction. 

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3. The extended generalized Haar-Walsh transform and applications

   https://doi.org/10.1117/12.2528923  

   Shao, Yiqun; Saito, Naoki (September 2019, Wavelets and Sparsity XVIII)

   Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important and much research has been done recently. Our previous Generalized Haar-Walsh Transform (GHWT) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh-Hadamard Transforms. This article proposes the extended Generalized Haar-Walsh Transform (eGHWT). The eGHWT and its associated best-basis selection algorithm for graph signals will significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N) where N is the number of nodes of an input graph. While the previous GHWT/best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than 1.5^N possible bases, the eGHWT/best-basis algorithm can find a better one by searching through more than 0.618 ⋅ (1.84)^N possible bases. This article describes the details of the eGHWT/basis-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Keywords: Multiscale basis dictionaries, wavelets on graphs, graph signal processing, adapted time-frequency analysis, the best-basis algorithm 

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4. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (October 2023, Combinatorica)

   Abstract A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$ $P_5$-free graph with clique number$$\omega \ge 3$$ $\omega \ge 3$has chromatic number at most$$\omega ^{\log _2(\omega )}$$ ${\omega }^{{log}_2\left(\omega \right)}$. The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$ $(5/27)3^{\omega }$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$ $P_5$, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$ $P_5$-free graphs, and our result is an attempt to approach that. 

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5. Construction of a Dirichlet form on Metric Measure Spaces of Controlled Geometry

   https://doi.org/10.1007/s11118-024-10144-6  

   Butaev, Almaz; Luo, Liangbing; Shanmugalingam, Nageswari (June 2024, Potential Analysis)

   Abstract Given a compact doubling metric measure spaceXthat supports a 2-Poincaré inequality, we construct a Dirichlet form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$that is comparable to the upper gradient energy form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$. Our approach is based on the approximation ofXby a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$using the Dirichlet form on the graph. We show that the$$\Gamma $$ $\Gamma$-limit$$\mathcal {E}$$ $E$of this family of bilinear forms (by taking a subsequence) exists and that$$\mathcal {E}$$ $E$is a Dirichlet form onX. Properties of$$\mathcal {E}$$ $E$are established. Moreover, we prove that$$\mathcal {E}$$ $E$has the property of matching boundary values on a domain$$\Omega \subseteq X$$ $\Omega \subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form$$\mathcal {E}$$ $E$) on a domain inXwith a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects. 

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