Minimum size tree-decomposition

Top PDF Minimum size tree-decomposition:

Minimum Size Tree-Decompositions

Minimum Size Tree-Decompositions

the minimum width of a path-decomposition with length l [9] 1 . We have generalized the problem of minimum size path-decomposition presented in [9], and introduced the problem of minimum size tree-decomposition in a shorter version of this paper [13]. To the best of our knowledge, no other paper has dealt with the computation of tree- decompositions with minimum size before [13]. However, very recently, following the work in [13] and [9], Bodlaender et al. [6] have proposed exact subexponential time algorithms to solve the problems of minimum size tree-decomposition and minimum size path-decomposition for a fixed width k in 2 O(n/ log(n)) time and showed that the two problems cannot be solved in 2 o(n/ log(n)) time, assuming the Exponential Time
En savoir plus

34 En savoir plus

Size-Constrained Tree Decompositions

Size-Constrained Tree Decompositions

Fig. 14. The problem of computing s k , for k ≥ 4, seems more intricate already in the case of trees. Indeed, our polynomial- time algorithms to compute s k , k ≤ 3, in trees mainly rely on the fact that, for any tree T , there exists a minimum-size tree-decomposition of T with width at most 3, where each bag induces a connected subtree. This is unfortunately not true anymore in the case of tree-decomposition with width 5. As an example, consider the tree G (with 10 nodes) obtained from a star with three 3 leaves by subdividing twice each edge. See in Fig. 14(b). s 5 (G) = 2 and any
En savoir plus

24 En savoir plus

Minimum Size Tree-decompositions

Minimum Size Tree-decompositions

Abstract Tree-decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally de- pends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, prac- tical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algo- rithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.
En savoir plus

7 En savoir plus

Minimum Size Tree-Decompositions

Minimum Size Tree-Decompositions

tree-decompositions with optimal width. Moreover, in such case, the time-complexity of above-mentioned dynamic programming algorithms becomes dominated by the size of the tree-decompositions and, therefore, it becomes interesting to minimize it. In this extended abstract, we deal with the problem of computing tree-decompositions with minimum size. Obviously, if the width is not constrained, then the problem is trivial since there always exists a tree- decomposition of a graph with one bag (the full vertex-set). Hence, given a graph G and an integer k ≥ tw(G), we consider the problem of minimizing the size of a tree-decomposition of G with width at most k.
En savoir plus

5 En savoir plus

Organizing the atoms of the clique separator decomposition into an atom tree

Organizing the atoms of the clique separator decomposition into an atom tree

We then investigate how we can make use of the tree structure of the clique tree T of H to derive some improvements on the time complexity. We introduce a process which traverses T and converts it into the corresponding atom tree of G in O(n(n + t)) time, where t is the number of 2-pairs of H. (A 2-pair is a pair of non-adjacent vertices such that every chordless path from one vertex to the other is of length 2, so t ≤ m); this approach can probably be improved upon to yield a better time bound. We complement this result with an algorithm to compute a minimal triangulation in O(n(n + m)) time, improving the result from [38], which makes the complexity of computing an atom tree from the input graph in O(n(n + m)) time.
En savoir plus

24 En savoir plus

The size at reproduction of canopy tree species in central Africa

The size at reproduction of canopy tree species in central Africa

Reproductive size thresholds have been quantified for tropical tree species in Malaysia (Thomas 1996, Davies & Ashton 1999), Panama (Wright et al. 2005), and Dominica Island (Thomas et al. 2015). These thresholds were quantified based on diameter and expressed as a function of height using species-specific allometric equations between height and diameter (Thomas 1996, Wright et al. 2005, Thomas et al. 2015). Reproductive size thresholds vary greatly among species. For instance, reproductive diameter thresholds varied from 1 to 19 cm among 11 pioneer Macaranga species in Malaysia (Davies & Ashton 1999). Also, they were neg- atively correlated with seed mass and relative shade tolerance: The Macaranga species with the highest light requirements had a relatively large reproductive size (maximum height from 6 to 31 m). Light requirement and seed mass were, however, signifi- cantly correlated with maximum height, suggesting that these results may have been driven by the positive relationship between maximum height and reproductive diameter. Indeed, species that attained a large stature showed larger reproductive size thresholds than smaller-statured species (Thomas 2011). This pattern is con- sistent with a systematic cost of reproduction such that early reproduction reduces the resources available for growth after maturity and thus limits maximum tree size (Kozłowski 1992, Obeso 2002).
En savoir plus

12 En savoir plus

The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects

The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects

December 21, 2004 Abstract A new way is presented to define for minimum cost spanning tree (mcst-) games the irreducible core, which is introduced by Bird in 1976. The Bird core correspondence turns out to have interesting monotonicity and additivity properties and each stable cost monotonic allocation rule for mcst-problems is a selection of the Bird core correspondence. Using the additivity property an axiomatic characterization of the Bird core correspondence is obtained. Key-words: cost allocation, minimum cost spanning tree games, Bird core, cost monotonicity, cone additivity.
En savoir plus

22 En savoir plus

Efficient determination of the k most vital edges for the minimum spanning tree problem

Efficient determination of the k most vital edges for the minimum spanning tree problem

than the complexity of the algorithm proposed by Liang [8] using Pettie’s result [12]. However, given the fact that α(m, n) is always less than 4 in practice, the complexity of these two algorithms can be deemed as equivalent. Moreover, the complexity of our algorithm is better than that of Liang’s algorithm for general k. More interestingly, unlike any other algorithm, our algorithm has two specific useful features. First, it can also determine an optimal solution for i Most Vi- tal Edges MST , for each 1 ≤ i ≤ k, with the same time complexity. Second, it can be easily adapted to establish an implicit enumeration algorithm based on a branch and bound procedure. We also present in this paper a formulation by a mixed integer program to solve k Most Vital Edges MST. We implement and test all these proposed algorithms using, for the implicit enumeration algorithm, different branching and evaluation strategies. The results show that the implicit enumeration algorithm is much faster than the explicit enumeration algorithm as well as the resolution of the mixed integer program and its use of memory space can handle instances of significantly larger size. Moreover, we propose an ε-approximate algorithm.
En savoir plus

15 En savoir plus

On the minimum size of an identifying code over all orientations of a graph

On the minimum size of an identifying code over all orientations of a graph

Proof. Let C be an identifying code of an orientation D of G. We shall use the Discharging Method to prove that |C| > ∆+2 2 |V (G)|. This method was often use to get lower bounds on the size of identifying codes, in particular for infinite grids, see e.g. [2, 6, 7, 8, 9]. We give an initial charge of 1 to every vertex v. Hence the total charge is |V (G)|. Then every vertex distribute its charge uniformly to the vertices in its identifier. In other

24 En savoir plus

Sapling size influences shade tolerance ranking among southern boreal tree species

Sapling size influences shade tolerance ranking among southern boreal tree species

Methods      We developed and evaluated field-parameterized models of probability of mortality as a function of recent radial growth for seven species important in boreal and near boreal forests. The species evaluated were trembling aspen, paper birch, yellow birch, white spruce, mountain maple, balsam fir and eastern white cedar. For each species, we attempted to sample an equal number of individuals across different size classes. Size classes varied depending on species growth rate, with fast growing species sampled in larger size classes (Table 1). For example, aspen, a clonal species reproducing primarily from root suckers, exhibits rapid early growth, often attaining a height of up to 1 m in the first year. Aspen clonal connections, although maintained in smaller saplings, may disappear in larger individuals (DesRochers & Lieffers 2001). When species were not abundant, we sampled two size classes, whereas for abundant species we used three size classes ( Table 1).
En savoir plus

10 En savoir plus

Maximum Size of a Minimum Watching System and the Graphs Achieving the Bound

Maximum Size of a Minimum Watching System and the Graphs Achieving the Bound

2 and a g3 must affect g 3 only. In the first two cases in Figure 51, a g5 should have been taken when choosing T , or, as in the third case and as in the previous figure, we can add all the edges between g 2 and g 3 and obtain K 5 . So we are left with the case when there are two (or more) g3’s with b. v.’s linked only to α in T , see Figure 52(a). If in G there is the edge {α 0 , a}, {α 0 , b} or {α 0 , β}, then again a spanning tree with a g5 could have been chosen, and if there is the edge {α, a} or {α, b} and neither {α, c} nor {α, d}, we can add to T all the edges between g 2 and g 3 in order to obtain K 5 in a maximal graph. So the only possibility not ruled out yet is if there are the edges, say, {α, b} and {α, c} (more edges in G can only help). Then Figure 52(b) shows how to save (at least) one watcher, by locating two watchers at α.
En savoir plus

45 En savoir plus

Particle size distribution and estimated carbon flux across the Arabian Sea oxygen minimum zone

Particle size distribution and estimated carbon flux across the Arabian Sea oxygen minimum zone

5 Conclusions We observed strong vertical gradients in particle size distri- bution, biogeochemical and biological variables at the upper boundary of the OMZ. A gradient in particle concentration was also observed at the lower oxycline but less intense. Al- though many aspects of OMZ functioning still remain un- known, our results support earlier studies showing a strong layering of biological communities and processes. Our new results can be used to further discuss causal mechanisms. In the upper part of the OMZ core, the anaerobic microbial res- piration probably enhanced production and accumulation of observed particles < 100 µm but did not modify the calcu- lated particulate vertical flux. No specific vertical change of PSD > 100 µm was observed in the core of the OMZ sug- gesting that particulate flux transformation was low in that layer. At the lower oxycline of some stations, changes of abundances in both small and large particles classes are asso- ciated to zooplankton-enriched layers. There, the accumula- tion of large particles enhanced the calculated POC vertical flux to the bathy-pelagic zone of the ocean. Finally, the lack or low intensity of large particle remineralization in the core of the OMZ and possible particulate repackaging in the lower oxycline may further increase the ocean carbon sequestration in the OMZ of Arabian Sea relative to non-OMZ situations.
En savoir plus

18 En savoir plus

Localized Minimum Spanning Tree Based Multicast Routing with Energy-Efficient Guaranteed Delivery in Ad Hoc and Sensor Networks

Localized Minimum Spanning Tree Based Multicast Routing with Energy-Efficient Guaranteed Delivery in Ad Hoc and Sensor Networks

In this work, we use a minimum spanning tree (MST) as a multicast backbone in order decide when a message has to be split into multiple packets addressing each a destination subset. In addition, we use a MST based localized next hop selection scheme which considers energy consumption of sending a message to a neighbor v over the progress achieved thanks to this node. Since this scheme is just a greedy heuristic, the message might get stuck at nodes having no better neighbor. Thus, MSTEAM also makes use of a new multicast generalization of the face recovery mechanism which is described for the first time in this work to the best of our knowledge. One of the key aspects of our solution is that it highly fits wireless ad hoc and sensor networks since it is fully localized. Indeed, forwarding nodes need to construct local MST’s using only information on their 1-hop neighborhood, which may be obtained thanks to simple beacon messages. MSTEAM is also well-suited for constrained mobile devices, since a MST may be efficiently computed in time O(n log n). Moreover, our scheme is loop-free and always achieves delivery, as long as a path exists between the source node and the destinations. We provide a theoretical analysis proving this assertion, as well as some experimental results which demonstrate that MSTEAM is very energy-efficient and outperforms existing geographic multicast schemes.
En savoir plus

22 En savoir plus

Mathematical analysis of a size structured tree-grass competition model for savanna ecosystems

Mathematical analysis of a size structured tree-grass competition model for savanna ecosystems

The analytical study of the model reveals three possible equilibria excluding tree-grass coexis- tence (desert, grassland, forest) along with equilib- ria for which woody and grassy components show durable coexistence (i.e. savanna vegetation). The number of such equilibrium points depends on the function used to model the increase of fire intensity with grass biomass(see Remark 2); for our model, we can have at most three savanna equilibria. We identified four ecologically meaningful thresholds that defined in parameter space regions of monos- tability, bistability as in Accatino et al. 2010, De Michele et al. 2011 and tristability with respect to the equilibria. Tristability of equilibria may mean that shifts from one stable state to another may often be less spectacular that hypothesized from previous models and that scenarios of vegetation changes may be more complex.
En savoir plus

19 En savoir plus

Minimum-maximum

Minimum-maximum

Document n° 42, créée le 31/5/2003 - Mis à jour le 13/7/2007 1. Aire minimum d'une lunule On considère la figure suivante : (C) est un cercle de centre O et de rayon 1, [AB] est un diamètre. À partir d'un point M de [AB], tracer deux demi-cercles de diamètre [AM] et [MB] (voir figure ci- dessous).

16 En savoir plus

Sur les surfaces minimum

Sur les surfaces minimum

Extraits des procès-verbaux des séances / Société philomathique de Paris.. Paris :A.[r]

3 En savoir plus

Minimum géométrique

Minimum géométrique

Hypatie construit un pont pour aller de A en B.[r]

2 En savoir plus

A Three-step Decomposition Method for Solving the Minimum-Fuel Geostationary Station Keeping of Satellites Equipped with Electric Propulsion

A Three-step Decomposition Method for Solving the Minimum-Fuel Geostationary Station Keeping of Satellites Equipped with Electric Propulsion

In the reference [33], the two first steps of a decomposi- tion technique for solving the station keeping control prob- lem is presented. For the first step, an indirect method based on the application of the PMP with mixed control- state constraints is applied to solve a simplified optimal SK control problem, without considering the hard con- straints on the control law (thrust constraints such as la- tency between two bursts of the same thruster and no si- multaneous thrusting for instance). In a second step, a numerical approach is used to enforce all the thrust con- straints left apart at the first step, thanks to dedicated equivalence schemes. If the operational constraints are re- spected, the spacecraft trajectory is composed of thrusting arcs separated by coasting arcs. Thus, the control profile switches from a time interval where all thrusters are off to a time interval for which one thruster is on, and vice-versa. Therefore, the system can be viewed as a switched system composed by one subsystem per thruster and one subsys- tem describing the dynamics during coasting arcs. The reference [34] takes advantage of the method proposed by [35] consisting in computing the optimal switching times of switched systems thanks to a time change of coordinates. The idea of the proposed paper is hence to demonstrate the benefit of solving the station keeping problem with a three-step method. The two first steps are the ones of [33] and the third step optimizing the switching times be- tween coasting arcs and actuated arcs highly improves the fuel consumption. Therefore, the proposed contribution describes the three-step decomposition method in a uni- fied framework, allowing to illustrate the benefit of this three-step method on a real example given by our indus- trial partner and aerospace manufacturer Thales Alenia Space.
En savoir plus

15 En savoir plus

Un salaire minimum européen ?

Un salaire minimum européen ?

Mais, incontestablement, le paragraphe qui a retenu l’attention, y compris de la grande presse, vise les salaires. La proposition elle-même, pourtant, est prudente. Il n’est nullement question, comme on l’a parfois un peu hâtivement entendu, de créer un salaire minimum européen unique applicable dans toute l’Union, ni même d’adopter des règles impératives européennes à variation nationale. Plus modestement, la Commission entend, par le biais du dialogue social promouvoir un système de salaire équitable pays par pays. Pour tout dire, ces phrases seraient bien inoffensives si la communication n’avait pas été accompagnée, le même jour, d’un document de consultation à destination des partenaires sociaux européen, visant à interroger ceux-ci « sur une éventuelle action visant à relever les défis liés à un salaire minimum équitable » (C(2020) 83 final du 14 janvier 2020). Ce document, beaucoup plus détaillé, trace des pistes d’un incontestable intérêt.
En savoir plus

4 En savoir plus

A Branch-and-cut-and-price algorithm for the Stackelberg Minimum Spanning Tree Game

A Branch-and-cut-and-price algorithm for the Stackelberg Minimum Spanning Tree Game

Keywords: Stackelberg Games, Spanning Trees, Branch-and-cut-and-price. 1 Introduction The Stackelberg Minimum Spanning Tree Game (StackMST) is defined in terms of a undirected graph G = (V, E), with two sets of edges, blue B and red R (E = B ∪ R, B ∩ R = ∅) and fixed costs {c e ≥ 0 : e ∈ R} assigned to the red edges. Once the leader of the game defines prices {p e : e ∈ B} to the blue edges, the follower chooses a minimum weight spanning tree (MST) ( V, E T ) of G, at cost  e∈B∩E T p e +  e∈R∩E T c e . StackMST aims at finding

9 En savoir plus

Show all 3245 documents...