Local search is a powerful heuristic in optimization and computer science, the complexity of which has been studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a function $f : V \to \mathbb{R}$. The local search problem is to find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few queries to the oracle as possible. We show that if a graph $G$ admits a lazy, irreducible, and reversible Markov chain with stationary distribution $\pi$, then the randomized query complexity of local search on $G$ is $\Omega\left( \frac{\sqrt{n}}{t_{mix} \cdot \exp(3\sigma)}\right)$, where $t_{mix}$ is the mixing time of the chain and $\sigma = \max_{u,v \in V(G)} \frac{\pi(v)}{\pi(u)}.$ This theorem formally establishes a connection between the query complexity of local search and the mixing time of the fastest mixing Markov chain for the given graph. We also get several corollaries that lower bound the complexity as a function of the spectral gap, one of which slightly improves a result from prior work.

翻译：局部搜索是优化与计算机科学中一种强大的启发式方法，其在白盒和黑盒模型下的复杂性已被广泛研究。在黑盒模型中，我们给定图 $G = (V,E)$ 以及通过预言机访问的函数 $f : V \to \mathbb{R}$。局部搜索问题旨在找到局部最小值顶点 $v$，即对所有 $(u,v) \in E$ 满足 $f(v) \leq f(u)$，且要求尽可能减少对预言机的查询次数。我们证明，若图 $G$ 允许一个懒惰、不可约且可逆的马尔可夫链，其平稳分布为 $\pi$，则 $G$ 上局部搜索的随机查询复杂度为 $\Omega\left( \frac{\sqrt{n}}{t_{mix} \cdot \exp(3\sigma)}\right)$，其中 $t_{mix}$ 为该链的混合时间，$\sigma = \max_{u,v \in V(G)} \frac{\pi(v)}{\pi(u)}$。该定理正式确立了局部搜索的查询复杂度与给定图最快混合马尔可夫链的混合时间之间的关联。我们进一步得到了若干推论，将复杂度下界表示为谱间隙的函数，其中一项推论对先前工作的结果略有改进。