New Algorithms for Weighted $k$-Domination and Total $k$-Domination Problems in Proper Interval Graphs

Chiarelli, Nina; Hartinger, Tatiana Romina; Leoni, Valeria Alejandra; Pujato, María Inés Lopez; Milanič, Martin

doi:10.48550/arxiv.1803.04327

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“…However, there is an algorithm that solves Distance-r Dominating Set on permutation graphs in time O(n 2 ) [16]; a much faster runtime. A concrete example of improving a mim-width based algorithm on a specific graph class has recently been provided by Chiarelli et al [8] who gave algorithms for the (Total) k-Dominating Set problems that run in time O(n 3k ) on proper interval graphs. The fastest previously known algorithm runs in time O(n 4+6k ) [3,6], the generic mim-width based algorithm (cf.…”

Section: Discussionmentioning

confidence: 99%

Generalized distance domination problems and their complexity on graphs of bounded mim-width

Jaffke¹,

Kwon²,

Strømme³

et al. 2018

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We generalize the family of (σ, ρ)-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-r dominating set and distance-r independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. To supplement these findings, we show that many classes of (distance) (σ, ρ)-problems are W[1]-hard parameterized by mim-width + solution size.

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