A polynomial algorithm for the k-cluster problem on interval graphs

This paper deals with the problem of finding, for a given graph and a given natural number k, a subgraph of k nodes with a maximum number of edges. This problem is known as the k-cluster problem and it is NP-hard on general graphs as well as on chordal graphs. In this paper, it is shown that the k-cluster problem is solvable in polynomial time on interval graphs. In particular, we present two polynomial time algorithms for the class of proper interval graphs and the class of general interval graphs, respectively. Both algorithms are based on a matrix representation for interval graphs. In contrast to representations used in most of the previous work, this matrix representation does not make use of the maximal cliques in the investigated graph.Comment: 12 pages, 5 figure

Computing and counting longest paths on circular-arc graphs in polynomial time.

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight

FAST IMPLEMENTATION TECHNIQUES OF MULTICHANNEL DIGITAL FILTERS FOR COLOR IMAGE PROCESSING USING MATRIX DECOMPOSITIONS

For the processing of color images, multivariable 3-input, 3-output 2-D digital filters are used, considering decomposition in the R, G and B components. Assuming that the three image components are decorrelated, three independent single-input, single-output (SISO) two-dimensional (2-D) digital filters are needed for the processing of each monochromatic image. Additional processing is needed for the correlated noise components in each chan- nel. The requirement of very fast processing dictates the use of special purpose hardware implementations. The VLSI array processors, which are special purpose, locally intercon- nected computing networks, are ideally suited for the fast implementation of digital filters, since they maximize concurrency by exploiting both parallelism and pipelining. In this paper fast implementation architectures of 3-input, 3-output 2-D multi-input digital filters for color image processing that are based on matrix decompositions are presented. The resulting structures are modular, regular, have high inherent parallelism and are easily pipelined, so that they may be implemented via VLSI array processors