AI 中文总结
该研究以值约束图的顶点覆盖数为结构参数，证明贪心与随机爬坡局部搜索可高效找到适应度景观的局部峰值，为局部搜索的复杂度分析提供了结构参数依据。
AI 中文摘要
人工智能领域中许多局部搜索方法可视为在对应离散适应度景观上的爬坡过程。从理论上讲，即使在这些适应度景观中找到局部峰值也是计算上难以处理的，但在实践中却常常可行。那么，适应度景观的哪些特征能实现高效的局部搜索？由于任何适应度景观都可由值约束的（超）图表示，本文将此问题重新表述为参数化复杂度问题：值约束图的何种结构参数能保证严格局部搜索可高效找到对应适应度景观中的局部峰值？对于顶点覆盖数为k的值约束图，本文证明，贪心局部搜索最多在2^(2k)·(n−k+1)步内找到局部适应度峰值，随机爬坡局部搜索最多在2^k·n(n−k)步内找到局部适应度峰值。本文还表明，这些结果对于严格局部搜索而言渐近性良好，因为存在顶点覆盖数为k的值约束图，从某些初始赋值出发的每次上升过程的长度至少为9/128·2^k·(n−k)。这表明顶点覆盖数是衡量局部搜索复杂度的良好结构参数。
英文摘要
Many local search methods for problems in artificial intelligence can be viewed as an uphill climb on a corresponding discrete fitness landscapes. Finding even local peaks in these fitness landscapes is computationally intractable in theory, but often works in practice. So what features of fitness landscapes allow for efficient local search? Since any fitness landscapes can be represented by a (hyper)graph of valued constraints, I re-frame this as a question of parameterized complexity: what structural parameter of a valued constraint graph guarantees that a strict local search will find a local peak in the corresponding fitness landscape efficiently? Given a valued constrain graph of vertex cover number k, I prove that greedy local search will find the a local fitness peak in at most $2^{2k}\cdot(n - k + 1)$ steps and random uphill local search will find a local fitness peak in an expected number of at most $2^k\cdot n(n - k)$ steps. I also show that these results are asymptotically good for strict local search because there are valued constraint graphs of vertex cover number $k$ where every ascent from some initial assignment has a length of $\frac{9}{128} \cdot 2^k \cdot (n - k)$ or greater. This suggests vertex cover number as a good structural parameter for the complexity of local search.
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