Kernels for Deletion to Classes of Acyclic Digraphs

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions

Hitting and Harvesting Pumpkins

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change

Παραμετρική επισκόπηση του Συνόλου Ανάδρασης Κορυφών

Στην παρούσα εργασία μελετώνται πυρήνες και παραμετρικοί αλγόριθμοι για το πρόβλημα Σύνολο Ανάδρασης Κορυφών τόσο για κατευθυνόμενα όσο και για μη κατευθυνόμενα γραφήματα. Στο πρόβλημα αυτό σκοπός μας είναι η διαγραφή το πολύ k κορυφών ενός γραφήματος G ώστε το γράφημα που θα προκύψει να είναι άκυκλο. Το πρόβλημα αυτό είναι NP-Hard, οπότε η έρευνα έχει στραφεί σε εναλλακτικούς τρόπους για την επίλυσή του (προσεγγιστικοί αλγόριθμοι, ευριστικές μέθοδοι, κ.ά.). Ένας από τους τρόπους αυτούς είναι με παραμετρικούς αλγόριθμους, στους οποίους επικεντρώνεται η παρούσα εργασία. Στα μη κατευθυνόμενα γραφήματα υπάρχει πυρήνας μεγέθους O(k^2) για την γενική περίπτωση. Στα κατευθυνόμενα γραφήματα είναι ανοιχτό πρόβλημα η ύπαρξη πυρήνα πολυωνυμικού μεγέθους για την γενική περίπτωση, ωστόσο στην παρούσα εργασία μελετάται η περίπτωση που η παράμετρος στο πρόβλημά μας είναι το Σύνολο Ανάδρασης Κορυφών για το αντίστοιχο μη κατευθυνόμενο γραφήμα, που έχει πυρήνα μεγέθους O(k^4). Όσον αφορά τους αλγόριθμους για τα μη κατευθυνόμενα γραφήματα, κυρίως χρησιμοποιούνται συνδυαστικά επιχειρήματα για την εύρεση της πολυπλοκότητάς τους, ενώ σε μια περίπτωση χρησιμοποιήκε πρόγραμμα σε Python για κάποιους υπολογισμούς. Για τα κατευθυνόμενα γραφήματα, χρησιμοποιούνται εργαλεία της θεωρίας γραφημάτων όπως οι τομές και οι διαχωριστές ενώ χρησιμοποιούνται και κάποιες πιο αφηρημένες δομές που ονομάζονται ε-δομές για την εύρεση συνόλου ανάδρασης. Η πολυπλοκότητα των αλγορίθμων αυτών προκύπτει πάλι με συνδυαστικά επιχειρήματα. Το Σύνολο Ανάδρασης Κορυφών είναι ένα εκτενώς μελετημένο πρόβλημα στην Παραμετρική Πολυπλοκότητα με πληθώρα εφαρμογών στα Λειτουργικά Συστήματα, στην κατασκευή κυκλωμάτων VLSI αλλά και στην εύρεση γονιδίων που ευθύνονται για τον καρκίνο και ανήκει σε μια μεγαλύτερη κατηγορία προβλημάτων που αφορούν Σύνολα Ανάδρασης. Παρόμοιο πρόβλημα είναι το Σύνολο Ανάδρασης Τόξων το οποίο επίσης λύνεται με τους αλγόριθμους για το Σύνολο Ανάδρασης Κορυφών. Ο αναγνώστης αναμένεται να είναι εξοικειωμένος με βασικές έννοιες θεωρίας γραφημάτων, αλγορίθμων, υπολογιστικής και παραμετρικής πολυπλοκότητας.On this thesis we study kernels and parameterized algorithms for the Feedback Vertex Set problem for directed and undirected graphs. In this problem we want to delete at most k vertices from a graph G to make it acyclic. This problem is NP-Hard, so the research is focused on alternative ways of solving the problem (e.g. approximation algorithms, heuristics, etc.). Another way is via parameterized algorithms, and this is the scope of this thesis. On undirected graphs there is a kernel of size O(k^2) for a general instance. On directed graphs the existence of a polynomial kernel for general instances is an open problem. On this thesis we study the case when the problem is parameterized by the Feedback Vertex Set of the corresponding undirected graph, which has kernel of size O(k^4). The algorithms for the undirected graphs make use of combinatorial arguments to find their complexity, and there is a case which used a Python program to find the complexity of the algorithm. For directed graphs, the algorithms make use of tools from graph theory (e.g. cuts, separators) and in some cases they use some more abstract stuctures, such as ε-structures. The complexity of the algorithms for the directed feedback vertex set is based again on combinatorial arguments. The Feedback Vertex Set is a well studied problem in Parameterized Complexity with many applications on Operational Systems, VLSI design and in the search of genes that cause cancer and it belongs to a bigger class of problems that searches for Feedback Sets. Similar is the Feedback Art Set problem which we can also solve with the algorithms for the Feedback Vertex Set. The reader should be familiar with basic notions of graph theory, algorithms, computational and parameterized complexity

Evaluation of the ingestive behaviour of the dairy cow under two systems of rotation with slope

The ingestive behaviour of grazing animals is modulated by the vegetation characteristics, topography and the type of stocking method. This research was carried out in 2019, at the Rumipamba CADER-UCE. It aimed to evaluate the impact of two contrasting stocking methods of dairy cows grazing a pasture with an average of slope >8.5%. Four dairy cows were set to graze a 0.4 ha paddock for 5 days for continuous stocking methods, while for the electric fence methods the dairy cows were restricted to 0.2 ha and the fence was moved uphill every 3 hours, repeating this process four times a day. Cow were equipped with activity sensors for 12 h per day. The whole procedure was repeated 2 times after realizing an equalization cuts and both paddocks, a rest time of 30 days and a random reassignment of paddocks to one of the treatments. The cows showed a difference in terms of the percentage of grazing P=0.0072, being higher with the electric fence (55% of the measurement time). From rising-plate-meter estimates of available biomass along the grazing periods, we calculated despite similar forage allowances (electric fence = 48.06 kg DM/cow/d and continuous = 48.21 DM/cow/d) a higher forage intake was obtained in the electric fence treatment (17.5 kg DM/cow/d) compared the continuous stocking (15.7 kg DM/cow/d) (P=0.006). In terms of milk production animals grazing under the differences electrical fence stocking method tended (P=0.0985) to produce more milk (17.39 kg/d) than those grazing in the continuous system (15.16 kg/d) due to the influence of the slope (P=0.05), while for milk quality the protein content was higher for the electric fence (33.7 g/l) than the continuous method (30.5 g/l) (P=0.039). None of the other milk properties differed between methods (P>0.05)

Dipterocarps protected by Jering local wisdom in Jering Menduyung Nature Recreational Park, Bangka Island, Indonesia

Apart of the oil palm plantation expansion, the Jering Menduyung Nature Recreational Park has relatively diverse plants. The 3,538 ha park is located at the north west of Bangka Island, Indonesia. The minimum species-area curve was 0.82 ha which is just below Dalil conservation forest that is 1.2 ha, but it is much higher than measurements of several secondary forests in the Island that are 0.2 ha. The plot is inhabited by more than 50 plant species. Of 22 tree species, there are 40 individual poles with the average diameter of 15.3 cm, and 64 individual trees with the average diameter of 48.9 cm. The density of Dipterocarpus grandiflorus (Blanco) Blanco or kruing, is 20.7 individual/ha with the diameter ranges of 12.1 – 212.7 cm or with the average diameter of 69.0 cm. The relatively intact park is supported by the local wisdom of Jering tribe, one of indigenous tribes in the island. People has regulated in cutting trees especially in the cape. The conservation agency designates the park as one of the kruing propagules sources in the province. The growing oil palm plantation and the less adoption of local wisdom among the youth is a challenge to forest conservation in the province where tin mining activities have been the economic driver for decades. More socialization from the conservation agency and the involvement of university students in raising environmental awareness is important to be done