量子-LOCAL模型下树上LCL问题的超对数间隙结果
Superlogarithmic Gap Result for LCLs on Trees in Quantum-LOCAL
AI总结：
该研究证明树上LCL问题存在超对数间隙，即其确定性LOCAL算法与量子-LOCAL算法的求解轮数存在$O(\u03b7 n)$与$n^{\u03a9(1)}$的指数级差异。
AI中文摘要：
我们证明，在树上，任何可由依赖于$n^{o(1)}$的分布求解的局部可检查标号问题（LCL）$\u03a0$，也可由$O(\u03b7 n)$轮确定性LOCAL算法求解。该结果通过对输入树的耙-压缩式分解，以及在分解的组件上对有界依赖分布的局部模拟得到。作为推论，树上的任何LCL问题要么可由$O(\u03b7 n)$确定性LOCAL算法求解，要么需要量子-LOCAL算法执行$n^{\u03a9(1)}$轮才能求解。
英文摘要：
We show that, on trees, any locally checkable labeling problem (LCL) $Π$ 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 $n^{Ω(1)}$ rounds to solve by a quantum-LOCAL algorithm.
