Local search is a powerful heuristic in optimization and computer science, the complexity of which was 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 as possible. The query complexity is well understood on the grid and the hypercube, but much less is known beyond. We show the query complexity of local search on $d$-regular expanders with constant degree is $\Omega\left(\frac{\sqrt{n}}{\log{n}}\right)$, where $n$ is the number of vertices. This matches within a logarithmic factor the upper bound of $O(\sqrt{n})$ for constant degree graphs from Aldous (1983), implying that steepest descent with a warm start is an essentially optimal algorithm for expanders. The best lower bound known from prior work was $\Omega\left(\frac{\sqrt[8]{n}}{\log{n}}\right)$, shown by Santha and Szegedy (2004) for quantum and randomized algorithms. We obtain this result by considering a broader framework of graph features such as vertex congestion and separation number. We show that for each graph, the randomized query complexity of local search is $\Omega\left(\frac{n^{1.5}}{g}\right)$, where $g$ is the vertex congestion of the graph; and $\Omega\left(\sqrt[4]{\frac{s}{\Delta}}\right)$, where $s$ is the separation number and $\Delta$ is the maximum degree. For separation number the previous bound was $\Omega\left(\sqrt[8]{\frac{s}{\Delta}} /\log{n}\right)$, given by Santha and Szegedy for quantum and randomized algorithms. We also show a variant of the relational adversary method from Aaronson (2006), which is asymptotically at least as strong as the version in Aaronson (2006) for all randomized algorithms and strictly stronger for some problems.

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，其复杂度已在白盒与黑盒模型中得到研究。在黑盒模型中，给定图$G = (V,E)$及对函数$f : V \to \mathbb{R}$的预言机访问。局部搜索问题旨在用尽可能少的查询次数找到局部最小值顶点$v$，即对于所有$(u,v) \in E$，满足$f(v) \leq f(u)$。网格与超立方体上的查询复杂度已被充分理解，但其他图结构上的结果尚不充分。我们证明在常度数$d$-正则扩展图上，局部搜索的查询复杂度为$\Omega\left(\frac{\sqrt{n}}{\log{n}}\right)$，其中$n$为顶点数。该结果在$O(\sqrt{n})$的上界（针对常度数图，由Aldous（1983）提出）的对数因子内达到匹配，表明带热启动的最速下降法是扩展图上的近似最优算法。此前已知的最优下界为$\Omega\left(\frac{\sqrt[8]{n}}{\log{n}}\right)$，由Santha与Szegedy（2004）对量子与随机算法证明。我们通过引入更广泛的图特征（如顶点拥塞度与分离数）框架得到该结果。证明表明：对任意图，局部搜索的随机查询复杂度为$\Omega\left(\frac{n^{1.5}}{g}\right)$，其中$g$为图的顶点拥塞度；以及$\Omega\left(\sqrt[4]{\frac{s}{\Delta}}\right)$，其中$s$为分离数，$\Delta$为最大度。针对分离数，此前的下界为$\Omega\left(\sqrt[8]{\frac{s}{\Delta}} /\log{n}\right)$，由Santha与Szegedy对量子与随机算法给出。我们还展示了Aaronson（2006）中关系对抗方法的一个变体，该变体对所有随机算法渐近不弱于Aaronson（2006）的原版本，且对某些问题严格更强。