Efficient enumeration of solutions produced by closure operations

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of the same name which appeared in STACS 2016. Final version for DMTCS journa

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio

On maximal chain subgraphs and covers of bipartite graphs

In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem. The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time

Incremental complexity of a bi-objective hypergraph transversal problem

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity

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

An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal hitting sets in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was open for a while.Comment: 13 pages, 1 figure, submitte

An average study of hypergraphs and their minimal transversals

International audienceIn this paper, we study some average properties of hypergraphs and the average com-plexity of algorithms applied to hypergraphs under different probabilistic models. Our approach is both theoretical and experimental since our goal is to obtain a random model that is able to capture the real-data complexity. Starting from a model that generalizes the Erdös-Renyi model [9, 10], we obtain asymptotic estimations on the average number of transversals, minimals and minimal transversals in a random hy-pergraph. We use those results to obtain an upper bound on the average complexity of algorithms to generate the minimal transversals of an hypergraph. Then we make our random model more complex in order bring it closer to real-data and identify cases where the average number of minimal tranversals is at most polynomial, quasi-polynomial or exponential

Canada v. Recours aux paradis fiscaux/bancaires : dans quelle mesure la politique de lutte du Canada peut-elle être améliorée?

Ce mémoire porte sur la lutte aux paradis fiscaux et bancaires et plus particulièrement sur les moyens de lutte dont s'est doté le Canada. Le mémoire est divisé en deux grandes parties. La première est consacrée aux différents instruments mis en place par le Canada ainsi qu'à divers problèmes rencontrés par cette juridiction dans cette lutte. Cette partie sera, dans un premier temps, descriptive. Dans un second temps, plusieurs propositions d'améliorations du système canadien seront émises. La deuxième partie analysera divers instruments de lutte, instaurés par d'autres juridictions, qui méritent d'être étudiées par le Canada en vue d'une éventuelle implementation en droit fiscal canadien. Nous nous pencherons ainsi, dans le présent mémoire, sur des thèmes tels que la conclusion de conventions préventives de double imposition avec des juridictions généralement qualifiées de paradis fiscaux ou bancaires, la double déductibilité des intérêts, les programmes de divulgation volontaire, etc

Making sense of a cophylogeny output: Efficient listing of representative reconciliations

4sìopenCophylogeny reconciliation is a powerful method for analyzing host-parasite (or host-symbiont) co-evolution. It models co-evolution as an optimization problem where the set of all optimal solutions may represent different biological scenarios which thus need to be analyzed separately. Despite the significant research done in the area, few approaches have addressed the problem of helping the biologist deal with the often huge space of optimal solutions. In this paper, we propose a new approach to tackle this problem. We introduce three different criteria under which two solutions may be considered biologically equivalent, and then we propose polynomial-delay algorithms that enumerate only one representative per equivalence class (without listing all the solutions). Our results are of both theoretical and practical importance. Indeed, as shown by the experiments, we are able to significantly reduce the space of optimal solutions while still maintaining important biological information about the whole space.openWang Y.; Mary A.; Sagot M.-F.; Sinaimeri B.Wang, Y.; Mary, A.; Sagot, M. -F.; Sinaimeri, B

Efficient enumeration of maximal split subgraphs and sub-cographs and related classes

In this paper, we are interested in algorithms that take in input an arbitrary graph $G$, and that enumerate in output all the (inclusion-wise) maximal "subgraphs" of $G$ which fulfil a given property $\Pi$. All over this paper, we study several different properties $\Pi$, and the notion of subgraph under consideration (induced or not) will vary from a result to another. More precisely, we present efficient algorithms to list all maximal split subgraphs, sub-cographs and some subclasses of cographs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal independent sets of an auxiliary graph. For cographs and some subclasses , the algorithms rely on a framework recently introduced by Conte & Uno called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property $\Pi$ that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every "interesting" hereditary property $\Pi$. We extend the hardness result to some specific edge version of the extension problem

The Perfect Matching Reconfiguration Problem

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time