# Weighted Total Acquisition

1 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
2 G-SCOP_ROSP [2020-....] - Recherche Opérationnelle pour les Systèmes de Production [2020-....]
G-SCOP [2020-....] - Laboratoire des sciences pour la conception, l'optimisation et la production [2020-....]
Abstract : In the Weighted Total Acquisition problem (WTA) on a weighted graph $G$ (only positive weights), a vertex $v$ can acquire the total weight of a neighbour $u$ if and only if the current weight of $v$ is at least that of $u$. The new weight of $u$ is then zero, and the new weight of $v$ is then the sum of the weights at $u$ and $v$ just before the acquisition. Over all possible acquisition sequences in $G$ with weight function $w$, the minimum number of vertices with a non-zero weight at the end is denoted by $a_t(G,w)$. Given a graph $G$, a weighting $w$, and an integer $k\geq 1$, the WTA problem asks whether $a_t(G,w)\leq k$. The Binary (Unary resp.) WTA problem corresponds to the WTA problem when the weights are encoded in binary (unary resp.). We prove the Binary WTA problem is NP-complete in trees of bounded degree or depth, and wheels, but that it is in XP for trees and wheels when parameterized by the solution size. We prove Binary WTA is NP-complete in $K_{3,n}$, planar graphs of pathwidth 2, and unit interval graphs even when $k=1$, and in trivially perfect graphs when $k\geq 2$ (but polynomial-time solvable when $k=1$). We prove that Unary WTA is polynomial-time solvable in graphs of bounded treewidth and degree. When only the treewidth is bounded, this algorithm is quasi-polynomial, i.e., it runs in time $W^{O(\log W)}$, where $W$ is the sum of the weights of the vertices. Moreover, we show that Unary WTA is FPT in trees when parameterized by the maximum degree. However, we show that WTA is strongly NP-complete in trivially perfect graphs and split graphs, even when $k=1$ in the latter.
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https://hal.inria.fr/hal-02880093
Contributor : Fionn Mc Inerney <>
Submitted on : Wednesday, June 24, 2020 - 3:53:47 PM
Last modification on : Friday, July 17, 2020 - 2:32:01 PM

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weighted_total_acquisition.pdf
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• HAL Id : hal-02880093, version 1

### Citation

Guillaume Bagan, Valentin Gledel, Marc Heinrich, Fionn Mc Inerney. Weighted Total Acquisition. [Research Report] Aix Marseille Université. 2020. ⟨hal-02880093⟩

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