Symbolic regression for approximating graph geodetic number

Graph properties are certain attributes that could make the structure of the graph understandable. Occasionally, standard methods cannot work properly for calculating exact values of graph properties due to their huge computational complexity, especially for real-world graphs. In contrast, heuristics and metaheuristics are alternatives proved their ability to provide sufficient solutions in a reasonable time. Although in some cases, even heuristics are not efficient enough, where they need some not easily obtainable global information of the graph. The problem thus should be dealt in completely different way by trying to find features that related to the property and based on these data build a formula which can approximate the graph property. In this work, symbolic regression with an evolutionary algorithm called Cartesian Genetic Programming has been used to derive formulas capable to approximate the graph geodetic number which measures the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Finding the exact value of the geodetic number is known to be NP-hard for general graphs. The obtained formulas are tested on random and real-world graphs. It is demonstrated how various graph properties as training data can lead to diverse formulas with different accuracy. It is also investigated which training data are really related to each property

Exact Methods for the Longest Induced Cycle Problem

The longest induced (or chordless) cycle problem is a graph problem classified as NP-complete and involves the task of determining the largest possible subset of vertices within a graph in such a way that the induced subgraph forms a cycle. Within this paper, we present three integer linear programs specifically formulated to yield optimal solutions for this problem. The branch-and-cut algorithm has been used for two models. To demonstrate the computational efficiency of these methods, we utilize them on a range of real-world graphs as well as random graphs. Additionally, we conduct a comparative analysis against approaches previously proposed in the literature

The Fritz-John Condition System in Interval Branch and Bound method

The Interval Branch and Bound (IBB) method is a good choice when a rigorous solution is required. This method handles computational errors in the calculations. Few IBB implementations use the Fritz-John (FJ) optimality condition to eliminate non-optimal boxes in a constrained non-linear programming problem. Applying the FJ optimality condition implies solving an interval-valued system of equations. In the best case, the solution is an empty set if the interval box does not contain an optimizer point. Solving this system of equations is complicated or unsuccessful in many cases. This problem can be caused by the interval box being too wide, the defined system of equations containing unnecessary constraints, or the solver being unsuccessful. These unsuccessful attempts have a negative outcome and only increase the computation time. In this study, we propose some modifications to reduce the running time and computational requirements of the Interval Branch and Bound method

The Fritz-John Condition System in Interval Branch and Bound method

The Interval Branch and Bound (IBB) method is a good choice when a rigorous solution is required. This method handles computational errors in the calculations. Few IBB implementations use the Fritz-John (FJ) optimality condition to eliminate non-optimal boxes in a constrained non-linear programming problem. Applying the FJ optimality condition implies solving an interval-valued system of equations. In the best case, the solution is an empty set if the interval box does not contain an optimizer point. Solving this system of equations is complicated or unsuccessful in many cases. This problem can be caused by the interval box being too wide, the defined system of equations containing unnecessary constraints, or the solver being unsuccessful. These unsuccessful attempts have a negative outcome and only increase the computation time. In this study, we propose some modifications to reduce the running time and computational requirements of the Interval Branch and Bound method

Photoautotrophic picoplankton – a review on their occurrence, role and diversity in Lake Balaton

Occurrence of the smallest phototrophic microorganisms (photoautotrophic picoplankton, APP) in Lake Balaton was discovered in the early 1980s. This triggered a series of systematic studies on APP and resulted in the setting of a unique long-term picoplankton dataset. In this review, we intend to summarize the obtained results and to give a new insight on APP ecology and diversity in Lake Balaton. According to the results, APP dynamics depends on trophic state, temperature, nutrient, and light availability, as well as grazing pressure. APP abundance in Lake Balaton decreased to a low level (1–2 × 105 cells mL−1) as a consequence of decreasing nutrient supply (oligotrophication) during the past more than two decades, and followed a characteristic seasonal dynamics with higher abundance values from spring to autumn than in winter. Concomitantly, however, the APP contribution to both phytoplankton biomass and primary production increased (up to 70% and 40–50%, respectively) during oligotrophication. Regarding annual pattern, picocyanobacteria are dominant from spring to autumn, while in winter, picoeukaryotes are the most abundant, most likely due to the different light and temperature optima of these groups. Within picocyanobacteria, single cells and microcolonies were both observed with mid-summer dominance of the latter which correlated well with the density of cladocerans. Community-level chromatic adaptation (i.e., dominance of phycoerythrin- or phycocyanin-rich forms) of planktonic picocyanobacteria was also found as a function of underwater light quality. Sequence analysis studies of APP in Lake Balaton revealed that both picocyanobacteria and picoeukaryotes represent a diverse and dynamic community consisting several freshwater genotypes (picocyanobacteria: Synechococcus, Cyanobium; picoeukaryotes: Choricystis, Stichococcus, Mychonastes, Nannochloris, and Nannochloropsis)