Answer Set Programming (ASP) is a generic problem modeling and solving framework with a strong focus on knowledge representation and a rapid growth of industrial applications. So far, the study of complexity resulted in characterizing hardness and determining their sources, fine-grained insights in the form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Unfortunately, for the well-known parameter treewidth disjunctive programs require double-exponential runtime under reasonable complexity assumptions. This quickly becomes out of reach. We deal with the classification of structural parameters for disjunctive ASP on the program's rule structure (incidence graph). First, we provide a polynomial kernel to obtain single-exponential runtime in terms of vertex cover size, despite subset-minimization being not represented in the program's structure. Then we turn our attention to strictly better structural parameters between vertex cover size and treewidth. Here, we provide double-exponential lower bounds for the most prominent parameters in that range: treedepth, feedback vertex size, and cliquewidth. Based on this, we argue that unfortunately our options beyond vertex cover size are limited. Our results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.

翻译：回答集编程（ASP）是一种通用的问题建模与求解框架，专注于知识表示，且在工业应用中迅速增长。目前，复杂性研究已刻画了难解性及其根源，以二分法形式呈现的细粒度洞察，以及详细的参数化复杂性景观。不幸的是，对于众所周知的参数树宽而言，在合理的复杂性假设下，析取程序需要双指数级运行时间，这很快变得难以实现。本文处理了析取ASP程序规则结构（关联图）的结构参数分类问题。首先，我们提供了一个多项式核，从而在顶点覆盖大小参数下获得单指数级运行时间，尽管子集最小化并未在程序结构中体现。接着，我们将注意力转向严格优于顶点覆盖大小与树宽之间的结构参数。在此方向上，我们证明了该范围内最突出参数（树深、反馈顶点集大小与团宽）的双指数级下界。基于此，我们认为不幸的是顶点覆盖大小之外的可选方案非常有限。我们的结果提供了深入的难解性研究，依托于从正常程序到析取程序的新型归约，以复杂性增长换取指数级参数压缩。