Parametrized Complexity of Weak Odd Domination Problems

Given a graph $G=(V,E)$, a subset $B\subseteq V$ of vertices is a weak odd dominated (WOD) set if there exists $D \subseteq V {\setminus} B$ such that every vertex in $B$ has an odd number of neighbours in $D$. $\kappa(G)$ denotes the size of the largest WOD set, and $\kappa'(G)$ the size of the smallest non-WOD set. The maximum of $\kappa(G)$ and $|V|-\kappa'(G)$, denoted $\kappa_Q(G)$, plays a crucial role in quantum cryptography. In particular deciding, given a graph $G$ and $k>0$, whether $\kappa_Q(G)\le k$ is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems associated with the quantities $\kappa$, $\kappa'$ and $\kappa_Q$ are known to be NP-Complete. In this paper, we consider the approximation of these quantities and the parameterized complexity of the corresponding problems. We mainly prove the fixed-parameter intractability (W$[1]$-hardness) of these problems. Regarding the approximation, we show that $\kappa_Q$, $\kappa$ and $\kappa'$ admit a constant factor approximation algorithm, and that $\kappa$ and $\kappa'$ have no polynomial approximation scheme unless P=NP.Comment: 16 pages, 5 figure

Un modèle débit-durée-fréquence pour caractériser le régime d'étiage d'un bassin versant

La méthodologie débit-durée-fréquence (QdF), appliquée ces dernières années aux étiages, a permis de définir quatre modèles types recouvrant l'ensemble des rivières étudiées. L'identification de la typologie du site étudié et l'estimation de deux descripteurs hydrologiques locaux suffisent au modèle, dit de référence, pour en déduire les courbes QdF (1j ≤ d ≤ 30j) en débit moyen minimum sur l'année (VCNd) ou débit seuil minimum annuel (QCNd) non dépassé sur ces mêmes durées. S'il est relativement aisé de définir les descripteurs hydrologiques, il est plus difficile d'identifier, sans observation de débit, le modèle à prendre en compte. En reconsidérant avec plus de rationalité la démarche d'identification des typologies, et en particulier les distributions multidurées relatives à chaque bassin, il est apparu possible d'évoluer vers un modèle unique pouvant être calé en chaque site observé. Ce nouveau concept de modélisation repose sur la propriété d'affinité des distributions, relatives aux échantillons de valeurs de durées d. Par souci de continuité avec l'approche QdF à référence typologique, la loi statistique log-normale à deux paramètres a été choisie. Le modèle, dont la conceptualisation est indépendante de la loi statistique choisie, aura dans le cas de la loi log-normale trois paramètres à ajuster sur les VCNd échantillonnés. Trente six sous bassins du bassin hydrographique de la Moselle ont été étudiés. Le modèle développé pour les débit moyens VCNd peut être appliqué aux débits seuils QCNd en conservant le même jeu de paramètres, grâce à une relation observée entre débits moyens et débits seuils. Cette nouvelle modélisation rationalise l'approche antérieure basée sur la typologie d'écoulement de basses eaux des bassins versants.The flow-duration-frequency (QdF) concept, as applied in recent years to low flows, has made it possible to establish four reference models (GALEA et al., 1999a), corresponding to four typologies. The hydrological variables concerned are the minimum mean discharge of the year defined for various continuous durations d (1day ≤ d ≤ 30day), called VCNd, and the annual minimum threshold discharge not exceeded over these same durations, called QCNd, according to OBERLIN (1992). These QdF models allow a description of the temporal variability of low flows observed for a river basin, from a statistical point of view. The typology of the basin and two local hydrological descriptors have to be known. For ungauged basins, these two descriptors (GALEA et al., 1999b) are well estimated by various methods, such as multivariate analysis relating to the physiographic characteristics of the basin. Nevertheless, the choice of the reference model still remains contentious.By reconsidering in a more rational manner the step of identification of typologies, and in particular the discharge distributions (for durations d) relating to each basin, it appeared interesting to establish a local model. This new model has a simpler formulation, thanks to a scale invariance assumption. This research (CHAPUT, 1999) was undertaken on 36 sub-basins of the Mosel basin. In order to ensure continuity with the earlier QdF models described above, the two-parameter log-normal law was chosen and adjusted on the distribution of mean discharges. The scale invariance assumption is deduced from the observed parallelism of distributions related to different durations, when discharges are represented in a logarithm scale. This observation means that all of the distributions can be translated to a common point, in order to obtain one "consolidated" distribution, independent on the considered duration. This parallelism has been observed on many basins, and seems to be a realistic assumption. Furthermore, these observations have been made on samples, and do not depend on the choice of statistical law. The methodology described in this paper makes it possible to adjust the local QdF model on sampled discharges. Only three parameters have to be determined: sc, the "consolidated" standard deviation, ∆e the low flow characteristic duration and VCN(2,1), which represents the quantile of the one-day distribution, with the two-year return period (F=0.5).This model is also useful for the determination of threshold discharges (QCNd). An observed property gives a relation between the VCN and QCN quantiles, for a fixed return period, considering different durations d: VCN quantiles can been deduced from QCN quantiles by integrating them, according to d. Consequently, the analytical formulation of the VCN model can be derived according to d, in order to obtain a QCN model. This model has the same three parameters sc, ∆e and VCN(2,1) described above. The comparison between QCN quantiles adjusted on samples and QCN quantiles deduced from the VCN model by derivation shows good results.As a conclusion, this new modelling approach unifies the typological approach for both mean discharges and threshold discharges. It is based on a local adjustment and avoids having to choose between one of the four former reference models. This local model opens up perspectives for a regional model, as it has been done for floods, for example by the Group of Research in Statistical Hydrology (1996). This will make it possible to estimate the low flow regime on an ungauged basin

Modélisation régionale des débits de crue du bassin hydrographique du Cris : approche régionale classique et par modèles de référence

Une régionalisation débit-durée-fréquence des débits de crue est réalisée sur les sous bassins du Cris qui draine une superficie d'environ 14 300 km2 à l'ouest de la Roumanie. Cette régionalisation concerne 78 sous bassins dont les chroniques de débit quotidien et de pointe observées sont de trente ans en moyenne et pour lesquels nous disposons des pluies maximales de bassin de 1 jour à 10 jours calculées à partir de 92 postes pluviométriques. La régionalisation est menée selon deux approches : une approche régionale classique et une approche à partir de trois modèles adimensionnels de référence établis sur trois sites observés de France. La différence fondamentale entre les deux approches réside en ce que l'une prend en compte l'information spatiale pluie-débit inventoriée du Cris et que l'autre considère essentiellement l'information pluie-débit de chaque site de référence français. L'approche modèles de référence a pour base conceptuelle une typologie des crues qui pour un site cible est prédéfinie par un critère de choix, tandis que l'approche classique nécessite que soient définis des régions hydrologiques homogènes. Cette démarche est menée sur les trois sous bassins hydrographiques du Cris et permet d'étendre la région hydrologique homogène à l'ensemble du bassin du Cris. L'approche régionale comme l'approche modèles de référence privilégie la loi exponentielle adaptée aux valeurs supérieures à un seuil pour ce qui concerne les quantiles de crue de faible période de retour et pour des durées de 1j à 10j selon la dynamique de crue des sous bassins. Pour les quantiles de crue de grande période de retour les deux approches sous tendent le modèle du GRADEX, forme d'extrapolation des distributions observées par le gradex des pluies maximales. Quelle que soit l'approche de régionalisation, en tout site cible doivent être disponibles deux descripteurs de régime : le débit de pointe décennal Q10 et une durée caractéristique de crue D. Afin de comparer essentiellement l'incertitude des modélisations sur les quantiles de crue, D et Q10 sont connus et déduits des observations. Les résultats présentés montrent une bonne validité du modèle régional ajusté sur l'ensemble du Cris. Ceci indique que la zone étudiée est relativement bien homogène. Concernant les modèles de référence, leur critère de choix n'apparaît pas pertinent lorsqu'on s'intéresse aux faibles périodes de retour, mais se révèle significatif pour les fortes périodes de retour. Ce résultat est en grande partie dû à la méthode d'extrapolation appliquée. Celle ci est liée à la méthode du GRADEX et utilise l'information locale sur les gradex de pluie, comme cela est souvent le cas en France. Il est à noter que ces modèles de référence établis sur des chroniques de débit et pluie d'avant 1992 n'ont pas été réactualisés. L'exemple du bassin du Cris montre qu'ils n'en gardent pas moins un caractère opérationnel pour l'estimation des quantiles de crue de durée d (0 < d(j) < 10) et de période moyenne de retour T (5 < T(an) < 1000).When local information on streamflows is insufficient for estimating flood quantiles, a Regional Flood Frequency Analysis (RFFA) is usually carried out. Once homogeneous hydrological regions are defined, methods such as the "Index-Flood" method can be applied. In most cases, the variable understudy is the maximum peak flood (instantaneous or mean daily value, depending available data). However, the severity of a flood is not only defined by its peak, but also by its volume and duration.For this reason, Cemagref has developed for several years an approach which, in addition to the classic flood/frequency relationship, takes into account the notion of duration: the flood-duration-frequency approach (QdF). In a similar manner to the rainfall intensity-duration-frequency analysis, averaged discharges are computed over different fixed durations d. For each duration, a frequency distribution of maximum averaged discharges is studied. Finally, a continuous formulation is fitted, as a function of the return period (T) and the duration (d) over which discharges have been averaged. Rare quantiles are determined using rainfall frequency information, according the so-called "GRADEX" method ("aesthetic" version). The regionalization of the QdF approach leaded to define three sets of dimensionless QdF curves. The originality of the method is that each set was fitted to one unique basin, located in France, and corresponding to a distinct hydrological regime. A choice criterion, involving maximal rainfall distributions, determines which reference basin has to be considered. In this way, the QdF regionalization differs from the classic concepts mentioned in the first paragraph. The classic approach takes into account the whole streamflow-rainfall information available on a homogeneous hydrological region, while the QdF regionalization developed at Cemagref considers the streamflow-rainfall information available on the three French reference basins. The aim of this research is to apply classical regional concepts (homogeneous regions) to the QdF approach and to compare results with thus obtained using the three French reference basins. One interest of the study is that it has been conducted in Romania, ie, outside the area where the three reference basins are located.The case study is carried out on the Cris river sub-catchments which cover an area of about 14 300 km2, in the west of Romania. This regionalization concerns 78 sub-catchments having about thirty years of streamflow measurements (daily flow and instantaneous flood peaks). For validation purpose, two basin sets are constituted: a calibration set (54 basins) and a validation set (24 basins). Furthermore, maximal rainfalls over 1 day to 10 days are available for these basins, from 92 rain-gauge stations. First, regional QdF curves are deduced according the definition of homogeneous regions. Three regions are defined, corresponding to the three main sub-catchments of the Cris basins. After different tests, it is shown that the whole Cris basin can be considered as a unique homogeneous region. Then, the regionalization involving the three French reference basins is carried out. In both cases, methods are applied with the exponential law adjusted on peak over threshold values, for small return periods. Durations are ranging from 1 day to 10 days, according to the flood dynamic of the studied basins. For long return periods, both approaches use the GRADEX method, which extrapolates discharge distributions according to the rainfall distributions. Whatever the regional approach used, two descriptors have to be estimated for each target site, in order to unscale the dimensionless regional QdF curves. Theses two descriptors are the 10-year-return-period peak flood Q10 and a flood characteristic duration D. In order to compare the uncertainty of the two approaches, D and Q10 are local values, deduced from observations.The results presented shows a good validity of the regional model fitted to the whole Cris basin. This indicates that this region is quite homogeneous. Concerning the reference models, their choice criterion does not appear to be pertinent for small return periods, but becomes relevant for high return periods. In this case, estimations are comparable to thus provided using the whole regional information available on the Cris basin. This result is mainly due to the extrapolation method used. It is related to the GRADEX method and takes into account the local information about rainfall gradex, as it is often done in France. One point to be noticed is that these reference models have not been updated since they have been established in 1992. Despite that, the Cris basin example shows their operational ability for estimating flood quantiles of duration d (0 < d(day) < 10) and return period T (5 < T(year) < 1000). From an operational point of view, the three reference basins can be a valuable option. Indeed, a regional analysis is sometime difficult to carry out: data not available or too costly, time available for the study to short, etc. On the other hand, the three reference basins method is easier to apply and requires less data. However, uncertainties and hypothesis of the method should be kept in mind. In particular, we should be aware that high return period quantiles are given according the GRADEX method, ie using the frequency rainfall information. Consequently, the hypothesis of this method should be respected

L'approche débit-durée-fréquence : historique et avancées

La prévention du risque d'inondation nécessite une connaissance détaillée du régime hydrologique en crue du bassin étudié. Dans ce but, l'approche débit-durée-fréquence (QdF), développée depuis déjà plusieurs années, a permis de définir un modèle statistique décrivant les crues observées en fonction de leur débit, de leur durée et de leur fréquence. Un récent travail a revisité cette approche. Grâce à son nombre réduit de paramètres, le modèle proposé, appelé modèle local convergent, peut être facilement ajusté pour chaque bassin. Dans l'ancienne approche, que nous appelons approche " bassin de référence ", l'ajustement local avait été effectué sur seulement trois bassins, dits de référence, et réputés être chacun représentatif d'une typologie d'écoulement différente. Ces trois paramétrisations types ont ensuite donné lieu à trois modèles adimensionnels, capables de caractériser la majorité des régimes observés. Le modèle adimensionnel correspondant au régime du bassin étudié devait être dénormé par deux caractéristiques locales du bassin : le débit instantané maximal de crue décennale et une durée caractéristique de crue. Une comparaison du nouveau modèle, appelé modèle local convergent, et de l'approche type " bassin de référence " a été effectuée sur une cinquantaine de bassins jaugés. Elle met en évidence la robustesse du modèle convergent et permet de discuter du choix du modèle relatif à l'approche " bassin de référence ". Le modèle local convergent autorise d'envisager le développement d'un modèle QdF régional, s'inspirant de différentes méthodes de régionalisation. Ceci permettra alors une application à des bassins peu ou non observés.Flood risk mitigation requires a good knowledge of hydrological flood regime, which can be described by a flow-duration-frequency (QdF) approach. New developments of this approach are presented and compared to the former method.Usually, flood frequency analysis deals only with the maximum flood peak distribution or the maximum daily discharge distribution. The QdF approach analyses maximum average flows over different durations d (d =1, 3, …, N days). Similar to intensity-duration-frequency curves, each of the QdF curves represents the flood frequency distribution, for the duration d. QdF modelling aims to express QdF curves by a Q(d,T) function (d : the duration; T : the return period).Before this present work, QdF modelling was associated with the "reference basin" approach. In this approach, QdF curves (plotted as a function of d, for fixed T) of many studied basins are converted into a dimensionless form. The two characteristics used are the 10-year peak flood, Q(d=0, T=10 years), and a characteristic flood duration (D) of the studied catchment, calculated from different flood hydrographs. Then, three different families are determined, grouping basins with similar dimensionless QdF curves. For each of these families, one reference basin is chosen. Their dimensionless curves are parameterised, in order to obtain a continuous formulation, as a function on T and d. By denormalising one of these dimensionless QdF models with the local parameters Q(0,10) and D, it is possible to obtain the continuous Q(d,T) formulation for the studied basin. The choice of the correct dimensionless model is made via a choice criterion. It involves Q(0,10), D and shape parameters of local maximal rainfall distributions (a Gumbel law is assumed), for different durations, d. These distributions are obtained according to the intensity-duration-frequency approach. If the studied basin is ungauged, local parameters Q(0,10) and D are estimated by regional formulas, involving significant variables such as catchment area and rainfall.Recent work has improved this "reference basin" approach. A new QdF model, called convergent local, has been developed. For fixed T, the model assumes that the Q(d,T) is described by a hyperbolic form, as a function of d. This choice of the hyperbolic form is based on the observation of many catchments (about one hundred). It has also been observed that QdF curves, plotted for fixed d as a function of T, converge toward the same point, when T decreases. Using these observations as assumptions, the model is then able to calculate Q(d,T) for any return period T and any duration d.If a two-parameter statistical law (such as the exponential law) is adopted, the model contains only 4 parameters. The first parameter is the limit of Q(d,T), when d tends to infinity. It is estimated by calculating the average value over the entire observed period of the Q(t) discharge time series. The second one gives the hyperbolas curvatures and is ∆. The ∆ parameter has a time dimension and is consequently a characteristic duration of the studied basin. The final two parameters are the location and shape parameters, x0 (0) and aq (0), of the exponential maximal flood distribution for d=0. x0 (0), aq (0) and ∆ parameters are directly adjusted on observed QdF curves of the studied basin.The comparison between the convergent local model and the "reference basin" approach has been carried out on about 50 basins, drawn from different regions of France. For each basin, the two approaches have been tested. First, the two characteristic durations D and∆, defined respectively by the "reference basins" approach and the convergent local model, are compared. As mentioned earlier, ∆ characteristic duration is an adjusted parameter and its calculation does not depend on D. In spite of their different definitions, a strong correlation between these two parameters is observed. This shows a good coherence between the two tested approaches. Second, in order to compare results, a relative mean error between calculated and observed values is determined for each basin and each model. Only the observed domain (T ≤ 20 years) has been considered, because the extrapolations cannot been validated with observed data.Concerning the "reference basin" approach, the three reference basin models are studied, and the choice criterion is applied. Results show that this choice criterion is not relevant. Concerning the convergent local model, the observed mean relative error is lower than in the "reference basin" approach. These good results are confirmed by a very small error dispersion. Consequently, the convergent local model is robust.As a conclusion, this paper presents new developments of the QdF approach: the convergent local continuous model. This model, locally adjusted, yields very satisfactory results. The next step is to apply it on ungauged basins, as is possible in the "reference basins" approach. This could be done by adapting regional methods, such as the index flood method

How crucial is it to account for the antecedent moisture conditions in flood forecasting? Comparison of event-based and continuous approaches on 178 catchments

This paper compares event-based and continuous hydrological modelling approaches for real-time forecasting of river flows. Both approaches are compared using a lumped hydrologic model (whose structure includes a soil moisture accounting (SMA) store and a routing store) on a data set of 178 French catchments. The main focus of this study was to investigate the actual impact of soil moisture initial conditions on the performance of flood forecasting models and the possible compensations with updating techniques. The rainfall-runoff model assimilation technique we used does not impact the SMA component of the model but only its routing part. Tests were made by running the SMA store continuously or on event basis, everything else being equal. The results show that the continuous approach remains the reference to ensure good forecasting performances. We show, however, that the possibility to assimilate the last observed flow considerably reduces the differences in performance. Last, we present a robust alternative to initialize the SMA store where continuous approaches are impossible because of data availability problems

Flash flood warning at ungauged locations: can proxy data be useful tocalibrate the hydrological model used?

International audienceThe Mediterranean region is subject to flash flood events that lead to many damages and sometimes to fatalities.Because not every little river’s tributaries can be monitored, flash floods occur most often on small ungaugedcatchments with limited available data. In this context, the calibration of the hydrological model used within aflood warning system is a tricky task.The aim of this study is to evaluate the potential benefit of ‘proxy data’ in areas where no classical continuousflow measurements are available. Administrative services for example collect and archive large amounts ofregional and local information over long periods based on observed damages, some of which can be used as proxyindicators for flood occurrences (e.g. infrastructure maintenance, insurance claims,...).To determine if such data could be helpful, a benchmark test is carried out, comparing three cases. In thefirst case the entire streamflow data series is used. In the second case, data from case 1 is degraded: only the dateand value of the flood peak are retained. In the last case, only the date of the flood peak is used. This last caseaims to imitate the case, where only proxy data collected by administrative services are available. The first casecorresponds to the classical gauged basin case, which is used as reference.These 3 cases are carried out using a simple conceptual hydrological model (from the GR model family)and data coming from around 150 catchments located in the South East of France and covering a 10-year period(1997-2007). The performance of simulated runoff is evaluated using contingency statistics (CSI). A split-sampletest is used in order to asses the robustness of the different calibrations.Results indicate that performances decrease from case 1 to case 3, case 2 beeing intermediate. Since themethodology developped for case 3 can be applied at ungauged locations, it seems that obtained alerts canbe very helpful when no classical hydrological data is available to calibrate the hydrological model used inthe flood warning system. This work has been made in the framework of the ongoing RHYTMME project.This project aims at developping a warning system in the south of French Alps, using radar rainfall data(http://rhytmme.cemagref.fr/synopsis)

Une approche géostatistique spatio-temporelle pour la prévision immédiate d'ensemble de pluies

3rd European Conference on Flood Risk Management, Lyon, FRA, 17-/10/2016 - 21/10/2016International audienceNowcasting systems are essential to prevent extreme events and reduce their socio-economic impacts. The major challenge of these systems is to capture high-risk situations in advance, with good accuracy, location and time. Uncertainties associated with the precipitation events have an impact on the hydrological forecasts, especially when it concerns localized flash flood events. Radar monitoring can help to detect the space-time evolution of rain fields, but nowcasting techniques are needed to go beyond the observation and provide scenarios of rainfall for the next hours of the event. In this study, we investigate a space-time geostatistical framework to generate multiple scenarios of future rainfall. The rainfall ensemble is generated based on space-time properties of precipitation fields given by radar measurements and rainfall data from rain gauges. The aim of this study is to investigate the potential of a framework that applies a geostatistical conditional simulation method to generate an ensemble nowcasting of rainfall fields. The Var region (south eastern France) and 14 events are used to validate the approach. Results show that the proposed method can be a solution to combine information from radar fields and rain gauges to generate nowcasting rainfall fields adapted for flash flood alert

Mise en oeuvre d'un outil d'alerte et de cartographie temps réel des aléas naturels liés aux précipitations dans les régions montagneuses et méditerranéennes du Sud-Est de la France

International audienceDue to its mountainous topography and its Mediterranean climate, the Provence-Alpes-Côte d'Azur (PACA) region in Southeastern France is particularly prone to flash floods, debris flows and mass movements (landslides and rockfall). A mapping system for these rainfall induced hazards has been tested by local and regional authorities and Government agencies since 2011 as part of the RHYTMME project. This system allows, thank to radar rainfall estimation and rainfall-runoff modelling, the real-time warning and monitoring of flash floods wherever they may occur in the PACA territory. It is also intended to enable, during intense rainfall events, the localisation of the streams susceptible to generate debris flows and of the slopes the more likely to trigger landslides and/or rockfalls

Comment passer d'un modèle hydrologique à un système de prévision des crues? Ecueils liés à la structure des modèles et aux échelles d'espace et de temps

Les modèles hydrologiques Pluie Débit sont des outils très utiles pour la prévision des crues. À l'heure actuelle, il n'est pas possible d'utiliser directement les modèles de simulation pour effectuer une bonne prévision. Nous explorons ici les différences entre modèles de simulation et modèles de prévision. Puis nous examinons l'importance relative des informations apportées au modèle : dans le passé, les forçages climatiques et les dernières observations de débit ; dans le futur, les prévisions de précipitations. La question des échelles spatiales est ensuite abordée et les limites d'une approche globale sont discutées dans une perspective opérationnelle. / Rainfall Runoff models are very useful tools for flood forecasting. As of today, the direct use of simulation models is not possible to get accurate predictions especially when it concerns short-term forecasting. In this paper, we explore the main differences between simulation and forecasting models. Then we assess the relative importance of every information provided to the model: the past climatic forcing and the last observed discharges; the future precipitation scenarios. Spatial scales are also examined and the limits of a global forecasting approach for operational purposes are discussed

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs