Tropical Dominating Sets in Vertex-Coloured Graphs

Bujtás, Csilla; Maftouhi, Hakim El; Karpiński, Marek; Manoussakis, Yannis; Montero, Leandro; Narayanan, Narayanan; Rosaz, Laurent; Thapper, Johan; Tuza, Źsolt

doi:10.1007/978-3-319-30139-6_2

Cited by 4 publications

(5 citation statements)

References 12 publications

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“… The decision problem RDC

_{1}

was known to be NP ‐complete, even for paths Anglès d'Auriac et al. (2016, 2018). We proved that it is NP ‐complete, even when restricted to trees where each color appears at most twice (Proposition 14).…”

Section: Discussionmentioning

confidence: 99%

“…Proof of Proposition In view of the subsequent proofs, we change slightly the proof from Anglès d'Auriac et al. (2016, 2018). The problem is clearly inside NP .…”

Section: Introductionmentioning

confidence: 99%

“…This work is motivated by Anglès d'Auriac et al. (2016, 2018), in particular their Theorem 2.1 (see Proposition 10) on the complexity of finding rainbow dominating sets in a vertex‐colored graph. Since locating‐dominating sets and identifying sets are particular classes of dominating sets, and since locating‐domination and identification are popular nowadays (see the ongoing bibliography at Lobstein, 2020), it seems natural to try to extend this theorem to these two classes, including when considering domination at distance

r

and uniqueness of solution.…”

Section: Introductionmentioning

confidence: 99%

“…When it is clear that we are dealing with a colored graph, we shall often drop the subscript col .Remark Here, we stick to the definition of a rainbow subset which is used in Anglès d'Auriac et al. (2016, 2018), because our article is intended to prolong and widen the complexity result therein. However, a different terminology can be found in the literature: a rainbow subset can also designate a vertex subset where each color appears at most once, whereas in a colorful subset, they appear exactly once; see, for example, Anglès d'Auriac et al.…”

Section: Introductionmentioning

confidence: 99%

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Some rainbow problems in graphs have complexity equivalent to satisfiability problems

Hudry

Lobstein

2020

Int Trans Operational Res

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In a vertex‐colored graph, a set of vertices S is said to be a rainbow set if every color in the graph appears exactly once in S. We investigate the complexities of various problems dealing with domination in vertex‐colored graphs (existence of rainbow dominating sets, of rainbow locating‐dominating sets, and of rainbow identifying sets), including when we ask for a unique solution: we show equivalence between these complexities and those of the well‐studied Boolean satisfiability problems.

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“… The decision problem RDC

_{1}

Section: Discussionmentioning

confidence: 99%

“…Proof of Proposition In view of the subsequent proofs, we change slightly the proof from Anglès d'Auriac et al. (2016, 2018). The problem is clearly inside NP .…”

Section: Introductionmentioning

confidence: 99%

r

and uniqueness of solution.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some rainbow problems in graphs have complexity equivalent to satisfiability problems

Hudry

Lobstein

2020

Int Trans Operational Res

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“…The problem, then, is equivalent to finding a set of vertex-disjoint cycles with the maximum total number of vertices in the cycles, and in the case of ties, finding a solution that secondarily contains the maximum number of colors. A subset of vertices from a vertex-colored graph is called tropical if it includes every color in the vertex-colored graph (Angles d'Auriac et al, 2016a). We define the decision and optimization versions of the TROPICAL-MAX-SIZE-EXCHANGE as follows.…”

Section: Problem 1b Max-size-exchangementioning

confidence: 99%

Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent

Highley,

2016

Preprint

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In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining whether it is possible to arrange an exchange of items among agents such that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGE is a similar problem, where the goal is to find a set of vertex-disjoint cycles that contains the maximum number of vertices and also contains all of the colors in the graph. We show that this problem is likewise NP-complete and APX-hard. For the restricted case where there are at most two vertices of each color (corresponding to a restriction that each agent may bring at most two items), both problems remain NP-hard but are in APX. Finally, we consider MAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include as many colors as possible and secondarily include as many vertices as possible. We show that this problem is NP-hard.

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