Knapsack with Vertex Cover, Set Cover, and Hitting Set

TL;DR

This work studies Knapsack problems constrained by graph-theoretic structures, notably Vertex Cover Knapsack and its variants, along with Set Cover Knapsack and $d$-Hitting Set Knapsack on hypergraphs. It establishes broad NP-hardness, including strong NP-completeness for several variants, and develops polynomial-time approximation schemes via primal-dual methods and LP relaxations, achieving $f$- and $H_d$-approximation factors for minimizing weight. The authors further show hardness of maximizing value and provide fixed-parameter tractable algorithms, including $(1-rac{1}{ ext{poly}})$-approximation results parameterized by treewidth, and specific $Oig(2^{ ext{tw}}ig)$- or $Oig(16^{ ext{tw}}ig)$-time dynamic programs for VC variants. Collectively, the results delineate the computational landscape of graph-constrained knapsack problems and yield practical, structure-exploiting algorithms for relevant instances.

Abstract

Given an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with vertex weights $(w(u))_{u\in\mathcal{V}}$, vertex values $(α(u))_{u\in\mathcal{V}}$, a knapsack size $s$, and a target value $d$, the \vcknapsack problem is to determine if there exists a subset $\mathcal{U}\subseteq\mathcal{V}$ of vertices such that $\mathcal{U}$ forms a vertex cover, $w(\mathcal{U})=\sum_{u\in\mathcal{U}} w(u) \le s$, and $α(\mathcal{U})=\sum_{u\in\mathcal{U}} α(u) \ge d$. In this paper, we closely study the \vcknapsack problem and its variations, such as \vcknapsackbudget, \minimalvcknapsack, and \minimumvcknapsack, for both general graphs and trees. We first prove that the \vcknapsack problem belongs to the complexity class \NPC and then study the complexity of the other variations. We generalize the problem to \setc and \hs versions and design polynomial time $H_g$-factor approximation algorithm for the \setckp problem and d-factor approximation algorithm for \hstp using primal dual method. We further show that \setcks and \hsmb are hard to approximate in polynomial time. Additionally, we develop a fixed parameter tractable algorithm running in time $8^{\mathcal{O}({\rm tw})}\cdot n\cdot {\sf min}\{s,d\}$ where ${\rm tw},s,d,n$ are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items for the \minimalvcknapsack problem.

Theorems & Definitions (19)

• Definition 1: Vertex Cover Knapsack
• Definition 2: Treewidth
• Definition 3: Set Cover Knapsack
• Definition 4: d- Hitting Set Knapsack
• Corollary 6
• Theorem 11
• Theorem 16
• Corollary 17
• Corollary 18
• Theorem 19