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的一种强大启发式算法，其复杂度已在白箱和黑箱模型中得到研究。在黑箱模型中，我们给定图 $G = (V,E)$ 以及函数 $f : V \to \mathbb{R}$ 的预言机访问权限。局部搜索问题旨在通过尽可能少的预言机查询，找到一个局部最小值顶点 $v$，即对所有 $(u,v) \in E$ 满足 $f(v) \leq f(u)$。我们证明：若图 $G$ 存在一个惰性、不可约且可逆的马尔可夫链，且其平稳分布为 $\pi$，则在 $G$ 上进行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 $t_{mix}$ 是该链的混合时间，$\sigma = \max_{u,v \in V(G)} \frac{\pi(v)}{\pi(u)}$。该定理正式建立了局部搜索的查询复杂度与给定图的最快混合马尔可夫链混合时间之间的联系。我们还得到了若干推论，以谱间隙的函数形式给出了复杂度的下界，其中一项略微改进了先前工作的结果。