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- W4387294574 abstract "Problems from metric graph theory such as Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact on the field of parameterized complexity by being the first problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. More specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to hypergraph dualization, arguably one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in different works this last decade: for which vertex (or edge) set graph property $Pi$ is the enumeration of minimal (or maximal) subsets satisfying $Pi$ equivalent to hypergraph dualization? As only very few properties are known to fit within this context -- namely, properties related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles of approach for tackling hypergraph dualization. In a second step, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show these cases to be mainly tractable." @default.
- W4387294574 created "2023-10-03" @default.
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- W4387294574 date "2023-09-29" @default.
- W4387294574 modified "2023-10-04" @default.
- W4387294574 title "Enumerating minimal solution sets for metric graph problems" @default.
- W4387294574 doi "https://doi.org/10.48550/arxiv.2309.17419" @default.
- W4387294574 hasPublicationYear "2023" @default.
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