Combinatorial explosion problem

The methods described in Section 12.2 can only locate the nearest minimum, which is normally a local minimum, when starting from a given set of variables. In some cases, the interest is in the lowest of all such minima, the global minimum in other cases it is important to sample a large (preferably representative) set of local minima. Considering that the number of minima typically grows exponentially with the number of variables, the global optimization problem is an extremely difficult task for a multidimensional function. It is often referred to as the multiple minima or combinatorial explosion problem in the Uterature. 

The relationship between linear superposition and quantum computing may be illustrated by electron spin which may be thought of as a binary system ( spin-up and spin-down ). Unlike informational tokens in classical digital computers which are either 0 or 1, both spin states coexist in the same state-space (i.e., are superposed) prior to an experimental measurement. Due to this situation (which cannot be visualized) 1 electron and 4 nearest neighbors in a weakly coupled system would exist in 2 or 32 configurations at one step of a computation. A biomolecular structure (e.g., a neural membrane) utilizing such parallelism could rapidly compute solutions to combinatorially explosive problems of physiology and behavior [31-33]. 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

Only one stereotype can be attached to an element. UML permits a stereotype to be defined as a combination of several others. This arrangement causes problems with combinatorial explosion as well as lack of semantics. 

It is obvious that the simultaneous inclusion of all possible reactor-reactive separation-separator design options into an automated design framework quickly leads to combinatorial explosion. In combination with the nonlinear models used to describe the reaction, mass, and heat transfer phenomena that occur in the processing units, the resulting synthesis problem is beyond the scope of existing optimization technology, even for relatively small problems. This has led to the... 

Full enumeration algorithms are confronted with a prevalent problem in computer science theory the naive assembly of molecular fragments to form virtual candidate compounds results in a combinatorial explosion. Owing to the enormous number of different molecular fragments and the way they can be... 

Our efforts have been directed toward the development of a new strategy for the construction of a CASE system that incorporates both treatments of the uncertainty inherent in the inverse problem. The uncertainties arise from the combinatorial explosion and the fuzzy character of the spectral information. It is clear that the process of structure elucidation must start from preliminary information considered more or less certain. This information is necessary to constrain the number of candidate structures within the solution space otherwise, this number will be infinite. In more general terms, each piece of information may serve as a constraint on the number of possible candidate structures. It is very... 

These kinds of algorithms make an attempt to explore all the degrees of freedom but face the problem of combinatorial explosion ... 

Another way to simplify the tertiary structure problem is to fix the backbone and then carry out an exhaustive search on the allowed side chain conformations. Desmet and co-workers have developed a dead-end elimination method for searching side chain conformations. Side chain conformations are grouped into a limited set of allowed rotamers. While an exhaustive search of all possible combinations of these rotamers is still not feasible, the application of the dead-end elimination theorem allows removal of impossible combinations early in the search, thus controlling the combinatorial explosion and leading to a small group of possible final solutions. The possible solutions can then be compared to find the best possible structure. 

With today s computational resources it is usually not a problem to exhaustively search compounds from corporate collections, vendor libraries, or small combinatorial libraries, which typically range in the order of 10 -10 molecules. However, for large virtual combinatorial libraries and collections thereof it becomes quickly unfeasible to enumerate all possible virtual molecules in advance due to combinatorial explosion. Consequently, there has been an increasing interest in computational methods to find alternative ways to systematically search large virtual combinatorial libraries, allowing a dramatic expansion of unexplored chemical space. 

It is desirable to reflect natural biodegradation diversity, but a complete set of all possible pathways can result in too many choices, a problem known as a combinatorial explosion . For example, if there were ten metabohtes at each stage of prediction and no convergence of metabolites, the munber of metabolites would increase by an exponent of ten at each step. The resultant thousands of pathways would be beyond hiunan evaluation on a reasonable time scale. To deal with this problem, it is necessary to further guide users by assigning priorities to every rule that governs each predicted reaction. 

Thus, to build a decision tree, one needs to explicitly enumerate all possible scenarios and the responses (decisions) to such scenarios. However, for some problems,. .., a combinatorial explosion of branches makes calculations cumbersome or impractical (Schuyler, 2001). One way that this problem is ameliorated (but not solved) is by introducing Monte Carlo simulations at each node of the decision tree. However, this does not address the problem of having to build the tree in the first place. In addition, trees are appropriate for the case where discrete decisions are made. Continuous decisions like for example the size of the investment, or more specifically, the size of a production plant, cannot be easily fit into decision trees without discretizing. 

Even a small molecular formula around C q produces a large number of possible chemical structures. The introduction of one heteroatom and/or a degree of unsaturation increases the size of the problem dramatically. The molecular formulae chosen in Tab. 23.1 represent comparably small compounds, far away from typical applications in modern organic chemistry. The main problem for making structure generation programs a common routine tool is the necessity to implement aU available pieces of information from the most important spectroscopic techniques at the earliest possible step in order to avoid the combinatorial explosion and therefore... 

---