On domination problems for permutation and other graphs

AbstractThere is an increasing interest in results on the influence of restricting NP-complete graph problems to special classes of perfect graphs as, e.g., permutation graphs. It was shown that several problems restricted to permutation graphs are solvable in polynomial time [2, 3, 4, 6, 7, 14, 16].In this paper we give 1.(i) an algorithm with time bound O(n2) for the weighted independent domination problem on permutation graphs (which is an improvement of the O(n3) solution given in [7]);2.(ii) a polynomial time solution for the weighted feedback vertex set problem on permutation graphs;3.(iii) an investigation of (weighted) dominating clique problems for several graph classes including an NP-completeness result for weakly triangulated graphs as well as polynomial time bounds

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research

Structural Completeness of a Multi-channel Linear System with Dependent Parameters

It is well known that the "fixed spectrum" {i.e., the set of fixed modes} of a multi-channel linear system plays a central role in the stabilization of such a system with decentralized control. A parameterized multi-channel linear system is said to be "structurally complete" if it has no fixed spectrum for almost all parameter values. Necessary and sufficient algebraic conditions are presented for a multi-channel linear system with dependent parameters to be structurally complete. An equivalent graphical condition is also given for a certain type of parameterization

On the Hardness and Inapproximability of Recognizing Wheeler Graphs

In recent years several compressed indexes based on variants of the Burrows-Wheeler transformation have been introduced. Some of these are used to index structures far more complex than a single string, as was originally done with the FM-index [Ferragina and Manzini, J. ACM 2005]. As such, there has been an increasing effort to better understand under which conditions such an indexing scheme is possible. This has led to the introduction of Wheeler graphs [Gagie et al., Theor. Comput. Sci., 2017]. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs, and that Wheeler graphs can be indexed in a way which is space efficient. Hence, being able to recognize whether a given graph is a Wheeler graph, or being able to approximate a given graph by a Wheeler graph, could have numerous applications in indexing. Here we resolve the open question of whether there exists an efficient algorithm for recognizing if a given graph is a Wheeler graph. We present: - The problem of recognizing whether a given graph G=(V,E) is a Wheeler graph is NP-complete for any edge label alphabet of size sigma >= 2, even when G is a DAG. This holds even on a restricted, subset of graphs called d-NFA\u27s for d >= 5. This is in contrast to recent results demonstrating the problem can be solved in polynomial time for d-NFA\u27s where d <= 2. We also show the recognition problem can be solved in linear time for sigma =1; - There exists an 2^{e log sigma + O(n + e)} time exact algorithm where n = |V| and e = |E|. This algorithm relies on graph isomorphism being computable in strictly sub-exponential time; - We define an optimization variant of the problem called Wheeler Graph Violation, abbreviated WGV, where the aim is to remove the minimum number of edges in order to obtain a Wheeler graph. We show WGV is APX-hard, even when G is a DAG, implying there exists a constant C >= 1 for which there is no C-approximation algorithm (unless P = NP). Also, conditioned on the Unique Games Conjecture, for all C >= 1, it is NP-hard to find a C-approximation; - We define the Wheeler Subgraph problem, abbreviated WS, where the aim is to find the largest subgraph which is a Wheeler Graph (the dual of the WGV). In contrast to WGV, we prove that the WS problem is in APX for sigma=O(1); The above findings suggest that most problems under this theme are computationally difficult. However, we identify a class of graphs for which the recognition problem is polynomial time solvable, raising the open question of which parameters determine this problem\u27s difficulty