Approximating Bandwidth
نویسنده
چکیده
is minimized. We denote B(G) = minf B(G, f) to be the bandwidth of G. This is equivalent to the matrix bandwidth minimization problem, which is the form it was introduced in the 1960s. In the matrix version of the problem, a symmetric matrix M is given, and the goal is to find a permuation matrix P such that the nonzero entries of PMP are all within a “band of minimum width” about the diagonal. An application of this is to reduce the amount of space required to store the matrix, and simplify some matrix operations such as the LU factorization. In the graph version, if we imagine the vertex ordering as a layout on a line, then the problem is to make the longest edge as short as possible. This has many engineering applications such as VLSI, where if the edges are wires, the problem becomes finding a layout such that the length of the longest wire is minimized, which is a factor in transmission delays. In terms of complexity, Papadimitriou [14] showed that the bandwidth minimization problem is NP-hard. Garey et al. [8] showed that the problem remains NP-hard even for trees with maximum degree 3. So this is one of a few graph theoretical problems that is hard even for trees. Subsequent investigations into this problem had been mainly focused on subclasses of graphs with specific structures, examples include caterpillars, chordal graphs, and even one on asteroidal triple-free claw-free graphs. The class of (generalized) caterpillars seems to be of some interest in the literature. A caterpillar is a graph that consists of a path P called the backbone, and several paths (called hairs or legs) that are attached (on one end) to vertices of the backbone. Clearly, caterpillars are trees. Assman et al. [1] gave a polynomial time algorithms for finding minimum bandwidth of caterpillars of hair length at most 2. However, that is the best that we can do for exact algorithms, since Monien [13] showed that for caterpillars of hair length at most 3, the problem is NP-hard.
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تاریخ انتشار 2005