We show that, on trees, any locally checkable labeling problem (LCL) $\Pi$ that can be solved by an $n^{o(1)}$-dependent distribution can also be solved by an $O(\log n)$-round deterministic LOCAL algorithm. The result is obtained through a rake-and-compress-style decomposition of the input tree, and local simulations of the bounded dependent distribution on the components of the decomposition. As a corollary to our result, any LCL problem on trees can either be solved by an $O(\log n)$ deterministic LOCAL algorithm, or requires $n^{\Omega(1)}$ rounds to solve by a quantum-LOCAL algorithm.
We present a distributed quantum algorithm that $3$-colors cycles in $O(1)$ rounds, with high probability. It follows that all locally checkable labeling problems (LCLs) that have round complexity $O(\log^* n)$ in the classical LOCAL model can be solved in $O(1)$ rounds in the quantum-LOCAL model, with high probability...
Maxime Flin, Long-Cheng Li, J. Suomela· 1 citation
It is shown that there are infinitely many locally checkable problems (i.e., problems whose solutions can be checked locally) whose round complexities belongs to the two intervals $[omega(1),o(\log\log^\star n)]$ and $[\omega(\log^\star n),o(\log n)] whenever these problems are considered in networks with unbounded max...
Filippo Casagrande, Pierre Fraigniaud, Benjamín Jauregui et al.· 0 citations
We present a quantum algorithm that generates a uniform superposition over the spanning trees of a graph -- also known as a q-sample -- using only a sub-linear number of queries to the graph. This goes beyond previous work that focuses on generating classical samples, including a recent quantum algorithm by Apers, Gao,...
Yassine Hamoudi, A. Tanasa, S. Sridhara· 0 citations
Gowers, Green, Manners, and Tao (Annals'25) recently resolved Marton's polynomial Freiman-Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace o...
We determine the $k$-colour size--Ramsey number of even cycles up to absolute constant factors. For every $k\ge2$ and every even $n\ge100\log k$, \[ \widehat R_k(C_n)=\Theta(k^2\log k)n. \] The lower bound follows from the corresponding result of Beke, Li and Sahasrabudhe for paths, while our upper bound improves the p...
This work settles this conjecture for parallel quantum algorithms, showing that every $t-query $d-round quantum algorithm can be simulated on most inputs with $t^{O(d^2)}$ classical queries, suggesting that for unstructured problems, superpolynomial speedups would require quantum circuits of superconstant depth, and ex...
Guy Blanc, Jordan Docter, Carmen Strassle et al.· 2 citations· ⚡1
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