In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).

Given a set P of paths over a graph, its edge intersection graph is a graph each of whose vertices corresponds to a path in P and where two vertices are connected if and only if the paths they correspond to share at least one edge. EPT is the class of edge intersection graphs of paths over trees. The complete characterization of EPT graphs by minimal forbidden subgraphs is not yet known. In this paper, we introduce two stretching operations on graphs and show that they preserve the property of being non-EPT graphs. We further investigate how to use these operations to construct minimal non- EPT graphs, allowing us to present one of the main results of this contribution, that is, two new infinite families of minimal non-EPT graphs that have not yet been described in the literature.

The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either V(G) \ C or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

Abstract The routing and spectrum assignment problem in modern optical networks is an NP‐hard problem that has received increasing attention during the last years. The majority of existing integer linear programming models for the problem uses edge‐path formulations where variables are associated with all possible routing paths so that the number of variables grows exponentially with the size of the instance. To bypass this difficulty, precomputed subsets of all possible paths per demand are typically used, which cannot guarantee optimality of the solutions in general. Our contribution is to provide a framework for the use of edge‐path formulations to minimize the spectrum width of a solution. For that, we select an appropriate subset of paths to operate on with the help of combinatorial properties in such a way that optimality of the solution can be guaranteed. Computational results indicate that our approach is indeed promising to solve the routing and spectrum assignment problem.

The routing and spectrum allocation (RSA) problem arises in the context of flexible grid optical networks, and consists in routing a set of demands through a network while simultaneously assigning a bandwidth to each demand, subject to non-overlapping constraints. One of the most effective integer programming formulations for RSA is the DR-AOV formulation, presented in a previous work. In this work we explore a relaxation of this formulation with a subset of variables from the original formulation, in order to identify valid inequalities that could be useful within a cutting-plane environment for tackling RSA. We present basic properties of this relaxed formulation, we identify several families of facet-inducing inequalities, and we show that they can be separated in polynomial time.

Time Expanded Networks, built by considering the vertices of a base network over some time space, are powerful tools for the formulation of problems that simultaneously involve resource assignment and scheduling. Still, in most cases, deriving algorithms from those formulations is difficult, due to both the size of the resulting models and the propagation of time rounding errors. The purpose of this paper is to address this algorithmic issue. We propose a generic Project and Lift decomposition scheme, and focus, inside this decomposition scheme, on the Lift issue, which consists in turning a solution defined on the base network into a solution in the time expanded space.

International audience

International audience

The routing and spectrum assignment problem is an NP-hard problem that receives increasing attention during the last years.Existing integer linear programming models for the problem are either very complex and suffer from tractability issues or are simplified and incomplete so that they can optimize only some objective functions.The majority of models uses edgepath formulations where variables are associated with all possible routing paths so that the number of variables grows exponentially with the size of the instance.An alternative is to use edgenode formulations that allow to devise compact models where the number of variables grows only polynomially with the size of the instance.However, all known edge-node formulations are incomplete as their feasible region is a superset of all feasible solutions of the problem and can, thus, handle only some objective functions.Our contribution is to provide the first complete edge-node formulation for the routing and spectrum assignment problem which leads to a tractable integer linear programming model.Indeed, computational results show that our complete model is competitive with incomplete models as we can solve instances of the RSA problem larger than instances known in the literature to optimality within reasonable time and w.r.t.several objective functions.We further devise some directions of future research.

VIPAFLEET is a framework to develop models and algorithms for managing a fleet of Individual Public Autonomous Vehicles (VIPA). We consider a homogeneous fleet of such vehicles distributed at specified stations in a closed site to supply internal transportation, where the vehicles can be used in different modes of circulation (tram mode, elevator mode, taxi mode). We treat in this paper a variant of the Online Pickup-and-Delivery Problem related to the taxi mode by means of multicommodity coupled flows in a time-expanded network and propose a corresponding integer linear programming formulation. This enables us to compute optimal offline solutions. However, to apply the well-known meta-strategy Replan to the online situation by solving a sequence of offline subproblems, the computation times turned out to be too long, so that we devise a heuristic approach h-Replan based on the flow formulation. Finally, we evaluate the performance of h-Replan in comparison with the optimal offline solution, both in terms of competitive analysis and computational experiments, showing that h-Replan computes reasonable solutions, so that it suits for the online situation.

This work is based on an extension of the Petri net framework. Our model relies on the definition of a priority relation between conflicting transitions, which is encoded in a compact manner by orienting the edges of a transition conflict graph. The benefit is that this allows the use of a successor function for the study of dynamic processes from a global point of view, independent from a particular initial state and the (complete) construction of the reachability graph. We address the problem of gaining the information that allows to provide an appropriate priority relation governing the dynamic behavior of the studied system and discuss some further implications and generalizations of the studied approach.

The context of this work is the reconstruction of Petri net models for biological systems from experimental data. Such methods aim at generating all network alternatives fitting the given data. For a successful reconstruction, the data need to satisfy two properties: reproducibility and monotonicity. In this paper, we focus on a necessary preprocessing step for a recent reconstruction approach. We test the data for reproducibility, provide a feasibility test to detect cases where the reconstruction from the given data may fail, and provide a strategy to cope with the infeasible cases. After having performed the preprocessing step, it is guaranteed that the (given or modified) data are appropriate as input for the main reconstruction algorithm.

The context of this work is the reconstruction of Petri net models for biological systems from experimental data. Such methods aim at generating all network alternatives fitting the given data. To keep the solution set small while guaranteeing its co

Abstract Building biological models by inferring functional dependencies from experimental data is an important issue in Molecular Biology. To relieve the biologist from this traditionally manual process, various approaches have been proposed to increase the degree of automation. However, available approaches often yield a single model only, rely on specific assumptions, and/or use dedicated, heuristic algorithms that are intolerant to changing circumstances or requirements in the view of the rapid progress made in Biotechnology. Our aim is to provide a declarative solution to the problem by appeal to Answer Set Programming (ASP) overcoming these difficulties. We build upon an existing approach to Automatic Network Reconstruction proposed by part of the authors. This approach has firm mathematical foundations and is well suited for ASP due to its combinatorial flavor providing a characterization of all models explaining a set of experiments. The usage of ASP has several benefits over the existing heuristic algorithms. First, it is declarative and thus transparent for biological experts. Second, it is elaboration tolerant and thus allows for an easy exploration and incorporation of biological constraints. Third, it allows for exploring the entire space of possible models. Finally, our approach offers an excellent performance, matching existing, special-purpose systems.

A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case $n\choose 2$ entries of the adjacency matrix of the given graph. Several graph properties have been shown to be elusive, e.g., planarity or k-colorability. A famous conjecture of Karp says that every nontrivial monotone graph property is elusive. We prove that a nonmonotone but hereditary graph property is elusive: perfectness.