Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs

The Transversal problem, i.e, the enumeration of all the minimal transversals of a hypergraph in output-polynomial time, i.e, in time polynomial in its size and the cumulated size of all its minimal transversals, is a fifty years old open problem, and up to now there are few examples of hypergraph classes where the problem is solved. A minimal dominating set in a graph is a subset of its vertex set that has a non empty intersection with the closed neighborhood of every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine, On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision 2014] that the enumeration of minimal dominating sets in graphs and the enumeration of minimal transversals in hypergraphs are two equivalent problems. Hoping this equivalence can help to get new insights in the Transversal problem, it is natural to look inside graph classes. It is proved independently and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal dominating sets in line graphs) can be enumerated in incremental output-polynomial time. We provide the first polynomial delay and polynomial space algorithm that lists all the minimal edge dominating sets in graphs, answering an open problem of [Golovach et al. - ICALP 2013]. Besides the result, we hope the used techniques that are a mix of a modification of the well-known Berge's algorithm and a strong use of the structure of line graphs, are of great interest and could be used to get new output-polynomial time algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Efficient Enumeration of Bipartite Subgraphs in Graphs

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensively studied, to the best of our knowledge. We propose the first non-trivial algorithms for enumerating all bipartite subgraphs in a given graph. As the main results, we develop two efficient algorithms: the one enumerates all bipartite induced subgraphs of a graph with degeneracy $k$ in $O(k)$ time per solution. The other enumerates all bipartite subgraphs in $O(1)$ time per solution

Extension of Some Edge Graph Problems: Standard and Parameterized Complexity

Le PDF est une version auteur non publiée.We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (resp., maximal) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (resp., a matching) and containing the forced edge set U (resp., avoiding any edges from the forbidden edge set E∖U). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation results

Role of Condom Negotiation on Condom use among Women of Reproductive Age in three Districts in Tanzania.

ABSTRACT: BACKGROUND: HIV/AIDS remains being a disease of great public health concern worldwide. In regions such as sub-Saharan Africa (SSA) where women are disproportionately infected with HIV, women are reportedly less likely capable of negotiating condom use. However, while knowledge of condom use for HIV prevention is extensive among men and women in many countries including Tanzania, evidence is limited about the role of condom negotiation on condom use among women in rural Tanzania. METHODS: Data originate from a cross-sectional survey of random households conducted in 2011 in Rufiji, Kilombero and Ulanga districts in Tanzania. The survey assessed health-seeking behaviour among women and children using a structured interviewer-administered questionnaire. A total of 2,614 women who were sexually experienced and aged 15--49 years were extracted from the main database for the current analysis. Linkage between condom negotiation and condom use at the last sexual intercourse was assessed using multivariate logistic regression. RESULTS: Prevalence of condom use at the last sexual intercourse was 22.2% overall, ranging from12.2% among married women to 54.9% among unmarried (single) women. Majority of the women (73.4%) reported being confident to negotiate condom use, and these women were significantly more likely than those who were not confident to have used a condom at the last sexual intercourse (OR = 3.13, 95% CI 2.22-4.41). This effect was controlled for marital status, age, education, religion, number of sexual partners, household wealth and knowledge of HIV prevention by condom use. CONCLUSION: Confidence to negotiate condom use is a significant predictor of actual condom use among women in rural Tanzania. Women especially unmarried ones or those in multiple partnerships should be empowered with condom negotiation skills to enhance their sexual and reproductive health outcomes

A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

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Graph Problems with Obligations

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Extension of Vertex Cover and Independent Set in Some Classes of Graphs

Lecture Notes in Computer Science book series (LNCS, volume 11485)International audienceWe study extension variants of the classical problems Vertex Cover and Independent Set. Given a graph G=(V,E) and a vertex set U⊆V, it is asked if there exists a minimal vertex cover (resp. maximal independent set) S with U⊆S (resp. U⊇S). Possibly contradicting intuition, these problems tend to be NP-complete, even in graph classes where the classical problem can be solved efficiently. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of theses problems, with parameter |U|, as well as the optimality of simple exact algorithms under ETH. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances. e further discuss the price of extension, measuring the distance of U to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability

Polynomial-Delay Enumeration of Maximal Common Subsequences

International audienceA Maximal Common Subsequence (MCS) between two strings X and Y is an inclusion-maximal subsequence of both X and Y. MCSs are a natural generalization of the classical concept of Longest Common Subsequence (LCS), which can be seen as a longest MCS. We study the problem of efficiently listing all the distinct MCSs between two strings. As discussed in the paper, this problem is algorithmically challenging as the same MCS cannot be listed multiple times: for example, dynamic programming [Fraser et al., CPM 1998] incurs in an exponential waste of time, and a recent algorithm for finding an MCS [Sakai, CPM 2018] does not seem to immediately extend to listing. We follow an alternative and novel graph-based approach, proposing the first output-sensitive algorithm for this problem: it takes polynomial time in n per MCS found, where n = max{|X|, |Y |}, with polynomial preprocessing time and space