Dominating Set Knapsack: Profit Optimization on Dominating Sets

TL;DR

This paper introduces the Dominating Set Knapsack problem, combining graph domination with profit maximization under budget constraints, and analyzes its computational complexity and algorithms for special cases.

Contribution

It formally defines the Dominating Set Knapsack problem, proves its complexity results, and develops FPT algorithms for specific graph classes and variants.

Findings

NP-complete on bipartite graphs

Weakly NP-complete on star graphs

Pseudo-polynomial algorithm for trees

Abstract

In a large-scale network, we want to choose some influential nodes to make a profit by paying some cost within a limited budget so that we do not have to spend more budget on some nodes adjacent to the chosen nodes; our problem is the graph-theoretic representation of it. We define our problem, Dominating Set Knapsack, by attaching the knapsack problem with the dominating set on graphs. Each vertex $v~(\in V) $ is associated with a cost factor $w(v)$ and a profit amount $\alpha(v)$. We aim to choose some vertices within a fixed budget $(s)$ that give maximum profit so that we do not need to choose their 1-hop neighbors. We show that the Dominating Set Knapsack problem is strongly NPC even when restricted to bipartite graphs, but weakly NPC for star graphs. We present a pseudo-polynomial time algorithm for trees in time $O(n\cdot min\{s^2, (\alpha(V))^2\})$. We show that Dominating Set Knapsack is unlikely to be Fixed Parameter Tractable (FPT) by proving that it is W[2]-hard parameterized by the solution size. We developed FPT algorithms with running time $O(4^{tw}\cdot n^{O(1)} min\{s^2,{\alpha(V)}^2\})$ and $O(2^{vck-1}\cdot n^{O(1)} min\{s^2,{\alpha(V)}^2\})$, where $tw$ represents the $tw$ of the given graph $G(V,E)$, $vck$ is the solution size of the Vertex Cover Knapsack, $s$ is the capacity or size of the knapsack and $\alpha(V)=\sum_{v\in V}\alpha(v)$. We obtained similar results for other variants $k-$Dominating Set Knapsack and Minimal Dominating Set Knapsack, where $k$ is the size of the dominating set.