Dominating Set Knapsack: Profit Optimization on Dominating Sets
Sipra Singh
TL;DR
Dominating Set Knapsack studies profit optimization under a knapsack budget with a domination constraint on graphs. The authors establish strong NP-hardness in general and bipartite graphs, while providing a pseudo-polynomial DP for trees. They place the problem in the parameterized complexity landscape, showing $\mathsf{W[3]}$-hardness for solution-size parameterization, but fixed-parameter tractability when parameterized by treewidth or Vertex Cover Knapsack size, with analogous results for related variants. The work highlights both fundamental intractability and practical tractability regimes, guiding algorithm design for network influence and resource allocation in graph-structured settings.
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.