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Optical Networks A Practical Perspective

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width.

https://drops.dagstuhl.de/opus/volltexte/2019/11138/pdf/LIPIcs-ESA-2019-17.pdf/

More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints

A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs

We resolve the longstanding open problem concerning the computational complexity of Max Cut on interval graphs by showing that it is NP-complete.

Complexity of Maximum Cut on Interval Graphs

A co-bipartite chain graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cardinality cut problem (\({\textsc {MaxCut}}\)) is \({\textsc {NP}}{\text {-hard}}\) in co-bipartite graphs (Bodlaender and Jansen, Nordic J Comput 7(2000):14–31, 2000). We consider \({\textsc {MaxCut}}\) in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that \({\textsc {MaxCut}}\) is polynomial time solvable in this graph class.

The maximum cardinality cut problem in co-bipartite chain graphs

Weighted upper domination number

Given a tree and a set \(\mathcal{P}\) of non-trivial simple paths on it, Vpt(\(\mathcal{P}\)) is the VPT graph (i.e. the vertex intersection graph) of \(\mathcal{P}\), and Ept(\(\mathcal{P}\)) is the EPT graph (i.e. the edge intersection graph) of the paths \(\mathcal{P}\) of the tree T. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). In this work, we define the graph Enpt(\(\mathcal{P}\)) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in \(\mathcal{P}\), and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths \(\mathcal{P}\) of T such that G = Ept \(\mathcal{P}\), and we say that 〈T, ,\(\mathcal{P}\)〉 is a representation of G. We show that trees, cycles and complete graphs are ENPT graphs. We characterize the representations of chordless ENPT cycles that satisfy a certain assumption. Unlike chordless EPT cycles which have a unique representation, these representations turn out to be multiple and have a more complex structure. Therefore, in order to give this characterization, we assume the EPT graph induced by the vertices of a chordless ENPT cycle is given, and we provide an algorithm that returns the unique representation of this EPT, ENPT pair of graphs. These representations turn out to have a more complex structure than chordless EPT cycles.

Graphs of Edge-Intersecting Non-splitting Paths in a Tree: Towards Hole Representations

The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and Non-Splitting Paths in a Tree (ENPT) 5. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph class by allowing the host graph to be an arbitrary graph. These are the graphs of Edge-Intersecting and Non-Splitting Paths ENP. Although the Edge Intersection Graphs of Paths in an arbitrary graph includes all graphs, we show that this is not the case for ENP. We also show that the class ENP coincides with the family of graphs of Edge-Intersecting and Non-Splitting Paths in a Grid (ENPG). Following similar studies for EPG graph class, we study the implications of restricting the number of bends of the individual paths in the grid. We show that such a restriction restricts the graph class. Specifically, by restricting the number of bends one gets an infinite sequence of classes such that every class is properly included in the next one.

/pdf/graphs-of-edge-intersecting-and-non-splitting-paths-14ja88xmht.pdf

Arman Boyacı

Papers

A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs

The maximum cardinality cut problem in co-bipartite chain graphs

Weighted upper domination number

Graphs of Edge-Intersecting Non-splitting Paths in a Tree: Towards Hole Representations

Graphs of edge-intersecting and non-splitting paths.