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 lower bound based on spectral gaps from prior work.

翻译：局部搜索是优化和计算机科学中一种强大的启发式方法，其复杂性已在白盒和黑盒模型中得到研究。在黑盒模型中，给定一个图$G = (V,E)$以及对函数$f : V \to \mathbb{R}$的预言访问。局部搜索问题旨在通过尽可能少的预言查询，找到一个局部最小顶点$v$，即对于所有$(u,v) \in E$满足$f(v) \leq f(u)$。我们证明，若图$G$允许一个具有平稳分布$\pi$的懒惰、不可约且可逆的马尔可夫链，则$G$上局部搜索的随机化查询复杂度为$\Omega\left( \frac{\sqrt{n}}{t_{mix} \cdot \exp(3\sigma)}\right)$，其中$t_{mix}$是该链的混合时间，且$\sigma = \max_{u,v \in V(G)} \frac{\pi(v)}{\pi(u)}$。该定理正式建立了局部搜索的查询复杂度与给定图上最快混合马尔可夫链的混合时间之间的联系。我们还得到了几个将复杂度下界表示为谱隙函数的推论，其中一个推论略微改进了先前工作中基于谱隙的下界。