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arXiv · 2506.24032

Dominating Set Knapsack: Profit Optimization on Dominating Sets

Sipra Singh
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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 $α(v)$. 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 $O(n\cdot min\{s^2, (α(V))^2\})$. 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 $O(4^{tw}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$ and $O(2^{vck-1}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$, 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 $α(V)=\sum_{v\in V}α(v)$. 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.

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Sipra Singh. 2026-01-29. Dominating Set Knapsack: Profit Optimization on Dominating Sets. https://arxiv.org/abs/2506.24032

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