A multivariate analysis of the strict terminal connection problem

Melo, Alexsander Andrade de; Figueiredo, Celina M. H. de; Souza, Uéverton S.

doi:10.1016/j.jcss.2020.02.001

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…It follows from the observations described in the previous paragraph that if, for some especial pair of vertices ρ 1 , ρ 2 ∈ R , the existence of the single edge ρ 1 ρ 2 ∈ E ( τ ( T ))\ E ( T ) is allowed, then the corresponding problem becomes NP‐complete. Contrasting with this fact, we prove that Connected router subgraph is polynomial‐time solvable for each fixed | R | ≥ 1, although it can be shown to be NP‐complete if | R | is not fixed (e.g., see Theorem 1 in [20]). Below, we present a formal definition for the problem.…”

Section: Connecting Terminals With Few Routersmentioning

confidence: 97%

“…More specifically, they showed that, for r ∈ {0, 1}, S‐TCP is Turing reducible to the problem of deciding whether a graph admits d vertex‐disjoint paths between a single given pair of vertices, whose sum of their lengths is at most a given positive integer x , which was proved to be polynomial‐time solvable by Suurballe [31]. In addition, Melo et al studied S‐TCP from the perspective of graph classes and parameterized complexity [20]. It was proved that S‐TCP restricted to split graphs can be solved in time

n^{O false(r false)}

but that the existence of an

f false(r false) \cdot n^{O false(1 false)}

‐time algorithm is unlikely for any computable function f , where n denotes the number of vertices of the input graph.…”

Section: Introductionmentioning

confidence: 99%

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On undirected two‐commodity integral flow, disjoint paths and strict terminal connection problems

2020

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Even, Itai, and Shamir (1976) proved simple two‐commodity integral flow is NP‐complete both in the directed and undirected cases. In particular, the directed case was shown to be NP‐complete even if one demand is unitary, which was improved by Fortune, Hopcroft and Wyllie (1980) who proved the problem is still NP‐complete if both demands are unitary. The undirected case, on the other hand, was proved by Robertson and Seymour (1995) to be polynomial‐time solvable if both demands are constant. Nevertheless, the complexity of the undirected case with exactly one constant demand has remained unknown. We close this 40‐year complexity gap, by showing the undirected case is NP‐complete even if exactly one demand is unitary. As a by product, we obtain the NP‐completeness of determining whether a graph contains 1 + d pairwise vertex‐disjoint paths, such that one path is between a given pair of vertices and d paths are between a second given pair of vertices. Additionally, we investigate the complexity of another related network design problem called strict terminal connection.

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Section: Connecting Terminals With Few Routersmentioning

confidence: 97%

n^{O false(r false)}

but that the existence of an

f false(r false) \cdot n^{O false(1 false)}

‐time algorithm is unlikely for any computable function f , where n denotes the number of vertices of the input graph.…”

Section: Introductionmentioning

confidence: 99%

On undirected two‐commodity integral flow, disjoint paths and strict terminal connection problems

2020

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“…The NP-completeness proof for TCP on chordal bipartite graphs were published in , and all the other mentioned results were published in [de Melo et al 2020, de Melo et al 2021a, de Melo et al 2023]. In particular, in [de Melo et al 2020], we analyse the complexity of S-TCP, the strict variant of TCP. This variant is also used as an auxiliary problem to solve TCP on split graphs.…”

Section: Connection Problemsmentioning

confidence: 99%

On (in)tractability of connection and cut problems

Melo

Figueiredo

Souza³

et al. 2023

Anais Do XXXVI Concurso De Teses E Dissertações (CTD 2023)

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This work addresses connection and cut problems from the viewpoint of graph classes and computational complexity, classic and parameterized. Regarding connection problems, we investigate the so-called TERMINAL CONNECTION problem (TCP), which can be seen as a generalisation of the classical STEINER TREE problem. We propose several complexity results for TCP, when restricted to specific graph classes, and some of its input parameters are fixed. As for cut problems, we analyse the complexity of the classical MAXCUT problem. We introduce the first complexity classification for the problem on interval graphs of bounded interval count. In addition, we prove the NP-completeness of MAXCUT on permutation graphs, settling a question posed by David S. Johnson in the Ongoing Guide to NP-completeness, which has been open since 1985. Finally, we consider the problem of computing the zig-zag number of a directed graph, which is a directed width measure defined over cuts of a graph.

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