Many combinatorial optimization problems can be formulated as finding an assignment that maximizes some pseudo-Boolean function (that we call the fitness function). Strict local search starts with some assignment and follows some update rule to proceed to an adjacent assignment of strictly higher fitness. This means that strict local search algorithms follow ascents in the fitness landscape of the pseudo-Boolean function. The complexity of the pseudo-Boolean function (and the fitness landscapes that it represents) can be parameterized by properties of the valued constraint satisfaction problem (VCSP) that encodes the pseudo-Boolean function. We focus on properties of the constraint graphs of the VCSP, with the intuition that spare graphs are less complex than dense ones. Specifically, we argue that pathwidth is the natural sparsity parameter for understanding limits on the power of strict local search. We show that prior constructions of sparse VCSPs where all ascents are exponentially long had pathwidth greater than or equal to four. We improve this this with our controlled doubling construction: a valued constraint satisfaction problem of pathwidth three where all ascents are exponentially long from a designated initial assignment. We conclude that all strict local search algorithms can be forced to take an exponential number of steps even on simple valued constraint graphs of pathwidth three.

翻译：许多组合优化问题可以表述为寻找一个赋值，以最大化某个伪布尔函数（称为适应度函数）。严格局部搜索从某个初始赋值开始，并遵循某种更新规则逐步转移到具有更高适应度的相邻赋值。这意味着严格局部搜索算法沿伪布尔函数适应度景观中的上升路径推进。伪布尔函数（及其所代表的适应度景观）的复杂度可以通过编码该伪布尔函数的值约束满足问题（VCSP）的性质来参数化。我们关注VCSP约束图的性质，其直观思想是：稀疏图比稠密图复杂度更低。具体而言，我们论证路径宽度是理解严格局部搜索能力极限的自然稀疏参数。我们证明了先前构造的、所有上升路径均为指数长度的稀疏VCSP，其路径宽度均大于等于四。通过我们提出的受控倍增构造法，我们改进了这一结果：构造了一个路径宽度为三的值约束满足问题，从指定的初始赋值出发，所有上升路径均为指数长度。我们得出结论：即使是在简单值约束图（路径宽度为三）上，所有严格局部搜索算法都可能被迫执行指数步长的操作。