Local search in combinatorial optimisation can be viewed as an uphill climb on a corresponding fitness landscape, where the assignments visited by a strict local search follow an ascent in the landscape. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by valued constraint satisfaction problems (VCSPs), it is natural to ask: what properties of VCSPs ensure that all ascents are polynomial? Or alternatively, what are the ``simplest'' VCSPs with exponential ascents? Prior examples of VCSPs with long ascents were built up as a chain of gadgets of constraints. Here we give a simpler star of gadgets construction by gluing 2n triangles of constraints at a common centre variable. We obtain a binary VCSP on 4n + 1 Boolean variables with an exponential ascent of length 10*2^n - 9. The variable at the centre of our construction intertwines two sublandscapes with only linear ascents into one with exponential ascents. The VCSP that we construct is significantly simpler than prior constructions in terms of treedepth (reducing Ω(log n) to 3) and feedback vertex set number (reducing Ω(n) to 1). We discuss the consequences of this simplicity for the parameterized complexity of local search.

翻译：组合优化中的局部搜索可被视为在相应适应度景观上的攀爬过程，其中严格局部搜索所访问的赋值遵循景观中的上升路径。这种爬山算法有时出奇地高效，但并非总是如此。由于适应度景观可通过带值约束满足问题简洁表示，自然引发以下问题：VCSP的哪些属性能保证所有上升路径均为多项式长度？或者反过来说，存在指数级上升路径的"最简单"VCSP是什么？此前具有长上升路径的VCSP示例是通过约束装置的链条结构构建的。本文给出一种更简单的星形装置构造——将2n个约束三角形粘合在一个公共中心变量上。我们获得了包含4n+1个布尔变量的二元VCSP，其具有长度为10*2^n - 9的指数级上升路径。该构造中的中心变量将两个仅存在线性上升路径的子景观交织成具有指数级上升路径的整体。相较于先前的构造，我们构建的VCSP在树深度（从Ω(log n)降至3）和反馈顶点集数（从Ω(n)降至1）方面显著简化。本文讨论了这种简化对局部搜索参数化复杂性的影响。