Many combinatorial optimization problems can be formulated as finding as assignment that maximized 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. From this, 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使某个伪布尔函数（我们称之为适应度函数）最大化的赋值。严格局部搜索从某个赋值开始，遵循某种更新规则转移到适应度严格更高的相邻赋值。这意味着严格局部搜索算法遵循伪布尔函数适应度景观中的上升路径。伪布尔函数（及其所表示的适应度景观）的复杂度可以通过编码该伪布尔函数的有值约束满足问题（VCSP）的性质进行参数化。我们重点关注VCSP约束图的性质，其直观理解是稀疏图比稠密图复杂度更低。具体而言，我们认为路径宽度是理解严格局部搜索能力极限的自然稀疏性参数。我们证明，先前构建的所有上升路径均呈指数级长度的稀疏VCSP实例，其路径宽度均大于或等于四。我们通过受控倍增构造改进了这一结果：构建了一个路径宽度为三的有值约束满足问题实例，其中从指定初始赋值出发的所有上升路径均具有指数级长度。由此我们得出结论：即使在路径宽度为三的简单有值约束图上，所有严格局部搜索算法也可能被迫执行指数级步数。