We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing $\ell$ vertex-disjoint objects of type X? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination X-HitPack can be significantly harder than these two base problems. Already for a particular choice of X, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain in the intersection of the areas of hitting and packing problems. On a high-level, we present two case studies: (1) X being all cycles, and (2) X being all copies of a fixed graph H. In each, we explore the classical complexity, as well as the parameterized complexity with the natural parameters k+l and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph with at least 3 vertices, the problem is \Sigma_2^P-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For example, for X being all cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.

翻译：我们研究了一类统一经典击中（也称为覆盖或横贯）与打包问题的一般化问题族。X-HitPack 实例询问：从图 G 中移除 k 个（可删除的）顶点是否能阻止我们打包 ℓ 个顶点不相交的 X 型对象？该问题涵盖了一系列问题，其两端分别为标准击中与打包问题。我们的主要动机问题是：X-HitPack 组合是否可能比这两个基础问题显著更难？即使对于特定的 X 选择，该问题也可针对多种复杂性概念提出，从而在击中与打包问题的交叉领域开创了一个迄今未探索的大领域。从高层次看，我们提出两个案例研究：（1）X 为所有环；（2）X 为固定图 H 的所有副本。在每个案例中，我们探索了经典复杂性，以及以自然参数 k+l 和树宽进行参数化的复杂性。我们观察到，组合问题可能比基础问题困难得多：对于环或 H 为至少包含 3 个顶点的连通图，该问题为 Σ_2^P-完全，且需要图树宽的双指数依赖（假设指数时间假设）。相比之下，在某些情况下，组合问题在运行时间上与基础问题定性相似，但需要显著的新思路。例如，对于 X 为所有环，我们通过一种涉及分支的方法建立了 2^poly(k+l)n^O(1) 算法。此外，对于 X 为所有边（即 H = K_2；这结合了顶点覆盖与最大匹配），该问题可在树宽为 tw 的图上于时间 2^\poly(tw)n^O(1) 内解决。实现该运行时间的关键步骤依赖于从匹配的代数（线性拟阵表示）表示获得的组合界。