Dominating sets reconfiguration under token sliding

Bonamy, Marthe; Dorbec, Paul; Ouvrard, Paul

doi:10.1016/j.dam.2021.05.014

Cited by 8 publications

(4 citation statements)

References 35 publications

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“…Finally, we consider what happens when we consider ℓ to be a parameter instead. Consider Timed Independent Set Reconfiguration (or equivalently Binary Timed Independent Set Reconfiguration) is parameterized by the size of the independent set and the length of the sequence 3 . Mouawad et al [17] showed that this problem is W [1]-hard 4 .…”

Section: Independent Set Reconfigurationmentioning

confidence: 99%

“…The dominating set reconfiguration problem is similar to the independent set reconfiguration problem, but in this case all sets in the sequence must form a dominating set in the graph. This again gives a PSPACE-complete problem [10], even for simple graph classes such as planar graphs and classes of bounded bandwidth [7], see also [3]. We define the parameterized problems Dominating Set Reconfiguration, Timed Dominating Set Reconfiguration and Binary Timed Dominating Set Reconfiguration similarly as their independent set counterparts, again parameterized by the number of tokens.…”

Section: Dominating Set Reconfigurationmentioning

confidence: 99%

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Parameterized Complexities of Dominating and Independent Set Reconfiguration

Bodlaender

Groenland

Swennenhuis

2022

Preprint

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We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves, XNL-complete when a maximum length l for the sequence is given in binary in the input, and XNLP-complete when l is given in unary. The problems were known to be W[1]- and W[2]-hard respectively when l is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.

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Section: Independent Set Reconfigurationmentioning

confidence: 99%

Section: Dominating Set Reconfigurationmentioning

confidence: 99%

Parameterized Complexities of Dominating and Independent Set Reconfiguration

Bodlaender

Groenland

Swennenhuis

2022

Preprint

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“…The rearrangement of atoms can be framed as a reconfiguration problem; the reconfiguration framework [2,3,5,[12][13][14][15]18] characterizes the transformation between configurations by means of a sequence of reconfiguration steps. By representing atoms as tokens, we can define each configuration of unlabelled tokens as a subset of vertices of a graph, indicating that there is a token placed on each vertex in the subset; for tokens with labels, a labelled configuration can be represented as a mapping of distinct labels to a subset of the vertices.…”

Section: Introductionmentioning

confidence: 99%

Parameterized complexity of reconfiguration of atoms

Cooper,

Maaz,

Mouawad

et al. 2023

Preprint

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Our work is motivated by the challenges presented in preparing arrays of atoms for use in quantum simulation. The recently-developed process of loading atoms into traps results in approximately half of the traps being filled. To consolidate the atoms so that they form a dense and regular arrangement, such as all locations in a grid, atoms are rearranged using moving optical tweezers. Time is of the essence, as the longer that the process takes and the more that atoms are moved, the higher the chance that atoms will be lost in the process.Viewed as a problem on graphs, we wish to solve the problem of reconfiguring one arrangement of tokens (representing atoms) to another using as few moves as possible. Because the problem is \textsf{NP}-complete on general graphs as well as on grids, we focus on the parameterized complexity for various parameters, considering both undirected and directed graphs, and tokens with and without labels. For unlabelled tokens, the problem is fixed-parameter tractable when parameterized by the number of tokens, the number of moves, or the number of moves plus the number of vertices without tokens in either the source or target configuration, but intractable when parameterized by the difference between the number of moves and the number of differences in the placement of tokens in the source and target configurations. When labels are added to tokens, however, most of the tractability results are replaced by hardness results.

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“…On the other hand, they gave linear-time algorithms for trees, interval graphs, and cographs. Bonamy et al [8] showed that DSR-TS is PSPACE-complete on split, bipartite and bounded treewidth graphs and polynomial-time solvable on dually chordal graphs and cographs. Bousquet and Joffard [13] showed that the problem is polynomial-time solvable on circulararc graphs and PSPACE-complete on circle graphs.…”

Section: Introductionmentioning

confidence: 99%