TY - GEN

T1 - Parameterized Complexity of Weighted Target Set Selection

AU - Suzuki, Takahiro

AU - Kimura, Kei

AU - Suzuki, Akira

AU - Tamura, Yuma

AU - Zhou, Xiao

N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024.

PY - 2024

Y1 - 2024

N2 - Consider a graph G where each vertex has a threshold. A vertex v in G is activated if the number of active vertices adjacent to v is at least as many as its threshold. A vertex subset A0 of G is a target set if eventually all vertices in G are activated by initially activating vertices of A0. The Target Set Selection problem (TSS) involves finding the smallest target set of G with vertex thresholds. This problem has already been extensively studied and is known to be NP-hard even for very restricted conditions. In this paper, we analyze TSS and its weighted variant, called the Weighted Target Set Selection problem (WTSS) from the perspective of parameterized complexity. Let k be the solution size and ℓ be the maximum threshold. We first show that TSS is W[1]-hard for split graphs when parameterized by k+ℓ, and W[2]-hard for cographs when parameterized by k. We also prove that WTSS is W[2]-hard for trivially perfect graphs when parameterized by k. On the other hand, we show that WTSS can be solved in O(nlogn) time for complete graphs. Additionally, we design FPT algorithms for WTSS when parameterized by nd+ℓ, tw+ℓ, ce, and vc, where nd is the neighborhood diversity, tw is the treewidth, ce is the cluster editing number, and vc is the vertex cover number of the input graph.

AB - Consider a graph G where each vertex has a threshold. A vertex v in G is activated if the number of active vertices adjacent to v is at least as many as its threshold. A vertex subset A0 of G is a target set if eventually all vertices in G are activated by initially activating vertices of A0. The Target Set Selection problem (TSS) involves finding the smallest target set of G with vertex thresholds. This problem has already been extensively studied and is known to be NP-hard even for very restricted conditions. In this paper, we analyze TSS and its weighted variant, called the Weighted Target Set Selection problem (WTSS) from the perspective of parameterized complexity. Let k be the solution size and ℓ be the maximum threshold. We first show that TSS is W[1]-hard for split graphs when parameterized by k+ℓ, and W[2]-hard for cographs when parameterized by k. We also prove that WTSS is W[2]-hard for trivially perfect graphs when parameterized by k. On the other hand, we show that WTSS can be solved in O(nlogn) time for complete graphs. Additionally, we design FPT algorithms for WTSS when parameterized by nd+ℓ, tw+ℓ, ce, and vc, where nd is the neighborhood diversity, tw is the treewidth, ce is the cluster editing number, and vc is the vertex cover number of the input graph.

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U2 - 10.1007/978-981-97-2340-9_27

DO - 10.1007/978-981-97-2340-9_27

M3 - Conference contribution

AN - SCOPUS:85193300725

SN - 9789819723393

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 320

EP - 331

BT - Theory and Applications of Models of Computation - 18th Annual Conference, TAMC 2024, Proceedings

A2 - Chen, Xujin

A2 - Li, Bo

PB - Springer Science and Business Media Deutschland GmbH

T2 - 18th Annual Conference on Theory and Applications of Models of Computation, TAMC 2024

Y2 - 13 May 2024 through 15 May 2024

ER -