Combinatorial optimization problems implicitly define fitness landscapes that combine the numeric structure of the 'fitness' function to be maximized with the combinatorial structure of which assignments are 'adjacent'. Local search starts at an assignment in this landscape and successively moves assignments until no further improvement is possible among the adjacent assignments. Classic analyses of local search algorithms have focused more on the question of effectiveness ("did we find a good solution?") and often implicitly assumed that there are no doubts about their efficiency ("did we find it quickly?"). But there are many reasons to doubt the efficiency of local search. Even if we focus on fitness landscapes on the hypercube that are single peaked on every subcube (i.e., semismooth fitness landscapes) where effectiveness is obvious, many local search algorithms are known to be inefficient. Since fitness landscapes are unwieldy exponentially large objects, we focus on their polynomial-sized representations by instances of valued constraint satisfaction problems (VCSP). We define a "direction" for valued constraints such that directed VCSPs generate semismooth fitness landscapes. We call VCSPs oriented if they do not have any pair of variables with arcs in both directions. Since recognizing if a VCSP-instance is directed or oriented is coNP-complete, we generalized oriented VCSPs as conditionally-smooth fitness landscapes that are recognizable in polynomial time for a VCSP-instance. We prove that many popular local search algorithms like random ascent, simulated annealing, history-based rules, jumping rules, and the Kernighan-Lin heuristic are very efficient on conditionally-smooth landscapes. But conditionally-smooth landscapes are still expressive enough so that algorithms like steepest ascent and random facet require a super-polynomial number of steps to find the fitness peak.

翻译：组合优化问题隐式地定义了适应度景观，该景观将待最大化的“适应度”函数的数值结构与“相邻”赋值之间的组合结构相结合。局部搜索从该景观中的一个赋值开始，依次移动赋值，直到相邻赋值中无法再找到改进解。经典的局部搜索算法分析更侧重于有效性（“我们是否找到了优质解？”）问题，并常常隐含地假设其效率（“我们是否快速找到了解？”）不存在疑问。然而，存在诸多理由质疑局部搜索的效率。即使我们关注超立方体上每个子立方体均为单峰（即半光滑适应度景观）的适应度景观——此时有效性显而易见——已知许多局部搜索算法仍是低效的。由于适应度景观是庞大且指数级规模的对象，我们通过值约束满足问题（VCSP）实例来关注其多项式规模的表示。我们为值约束定义“方向”，使得有向VCSP生成半光滑适应度景观。若一个VCSP不存在任何变量对在双向均存在弧，则称其为定向的。由于判定VCSP实例是否为有向或定向是coNP完全的，我们将定向VCSP推广为条件光滑适应度景观，该景观对于VCSP实例可在多项式时间内识别。我们证明，许多流行的局部搜索算法（如随机上升法、模拟退火、基于历史的规则、跳跃规则以及Kernighan-Lin启发式算法）在条件光滑景观上非常高效。然而，条件光滑景观仍具有足够的表达能力，使得诸如最陡上升法和随机面法等算法需要超多项式步数才能找到适应度峰值。