Hitting Meets Packing: How Hard Can it Be?

TL;DR

This work introduces X-HitPack, a unified framework that blends hitting and packing in graphs and analyzes the computational complexity across several natural choices of object type $\, ext{X}$. It establishes that Cycle-HitPack and connected-$H$-HitPack are $oldsymbol{ ext{ extsf{Sigma}}_2^ extsf{P}}$-complete, and that even on graphs of bounded treewidth the best possible running times can be double-exponential in the width (under ETH). The authors complement these hardness results with algorithmic results: a $2^{ ext{poly}(k+ olinebreak l)} olinebreak imes n^{O(1)}$-time algorithm for Cycle-HitPack, a $3^{k+l} olinebreak imes n^{O(1)}$-time algorithm for Edge-HitPack, and a $2^{ ext{poly}( ext{tw})} olinebreak imes n^{O(1)}$ algorithm for Edge-HitPack when $H=K_2$, with a broad suite of treewidth-based dynamic-programming techniques for more general $H$-HitPack cases. The paper also delivers several lower bounds: double-exponential lower bounds parameterized by pathwidth, and tight ETH-based separations, illustrating a rich landscape where the combined hitting/packing problems can be significantly harder than their individual components.

Abstract

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 Σ_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.

Figures (11)

• Figure 1.1: The areas of hitting and packing problems intersect in combinatorial duality and Erdős--Pósa property results, and in the algorithmic study of combined hitting and packing problems.
• Figure 2.1: If $H$ is not a clique (e.g., $H$ is a cycle on 5 vertices), then restricting a packing to $V_t$ may result in partial copies of $H$ that contain vertices from $V_t\setminus X_t$. Therefore, the description of a partial packing needs to include how these partial copies interact with $X_t$.
• Figure 2.2: An illustration of the construction for the double exponential lower bound for $H$-HitPack. The two topmost gadgets determine the clause number we consider. Each gadget $C_{\lambda,j,p}$ at the bottom encodes that literal $\lambda$ appears in the $j$th clause at position $p$. The hollow vertices indicate deleted vertices representing an assignment. The packing represented by the colored triangles represents the evaluation of the second clause that is $C_2 = (\overline x_1 \lor x_2 \lor x_3)$, which is not satisfied by the chosen assignment of $x_1=\mathsf{true}$ and $x_2=x_3=\mathsf{false}$.
• Figure 3.1: An illustration from the proof of \ref{['lem:intersect']}. The four highlighted paths at the bottom belong to the assumed maximal collection $\mathcal{P}$. For convenience, we use $i$ and $j$ to denote the neighbors of vertex $i,j\in F$ respectively. The nodes with diamond shape belong to $A=\bigcup_{P \in \mathcal{P}} A_P$, and after removing these nodes, there is one component which has more than one leaf (appearing in the box). Then, we can add the highlighted path in this box to $\mathcal{P}$ and thus, strictly increase the size of $\mathcal{P}$.
• Figure 7.1: An illustration for a node $t$ of how a partial packing and its type relate to each other. The black vertices and edges correspond to the edges and vertices of $G$. The deleted vertices are indicated by hollow dots. The highlighted vertices and edges show how the copies of $H$ are packed to the vertices of $G$. A highlighted vertex or edge with a white filling, indicates that we do not know to which vertices and edges of $G$ the ones of $H$ correspond.

Theorems & Definitions (144)

• Theorem 1
• proof
• proof : Proof of \ref{['lem:perfmatchbound']}
• proof : Proof of \ref{['thm:cycle-k-l']}
• Lemma 3.1
• Lemma 3.3
• proof
• Claim 3.4
• proof : Proof of Claim .
• Definition 3.5: Usable Paths and Cycle Packings