By Russell Schwartz
A practice-oriented survey of thoughts for computational modeling and simulation appropriate for a vast diversity of organic problems.
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Extra info for Biological Modeling and Simulation: A Survey of Practical Models, Algorithms, and Numerical Methods (Computational Molecular Biology)
Both maximum k-cut and minimum k-cut are NPcomplete. 3 Vertex Cover, Independent Set, and k-Clique Two related problems that often come up in practice are vertex cover and independent set. In the vertex cover problem, we want a set of edges of minimum size (or weight) such that every vertex in the graph has some edge in the cover incident on it. The formal decision problem for the unweighted version of vertex cover is the following: Vertex Cover Input An undirected graph G ¼ ðV ; EÞ and a bound B A R Question Does there exist a set of vertices V 0 J V such that for all ðu; vÞ A E, either u A V 0 or v A V 0 and jV 0 j a B?
To make such decisions, though, it helps to know as much as we can about common problems we are likely to encounter. We will now review some of the frequently occurring intractable problems and see what is known about alternatives for them. As with our quick examination of tractable problems, this chapter is far from exhaustive. It covers only a small selection of problems that are particularly likely to come up in biological modeling contexts. We will cover a set of graph problems and string problems, as well as some problems in set theory.
6 gives an example of the Steiner tree problem. Intraspecies phylogeny and similar problems in molecular evolution are some of the major motivations for studying Steiner tree problems. Often, as in intraspecies phylogeny, we are not explicitly given the Steiner nodes but must infer their existence. When our nodes are labeled with DNA bases, we may assume that any string of bases not in our input is a Steiner node in the graph. In such cases, we need the weight function to be speciﬁed in such a way that we can compute it between pairs of nodes not explicitly in the input.