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In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R 2 . First, we propose an O ( n 2 log n ) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangles, the time complexity can be slightly improved to O ( n log n + n K ) , where K is the size of the maximum clique. For disk graphs, we consider two variations of the maximum clique problem, namely geometric clique and graphical clique. The time complexity of our algorithm for finding the largest geometric clique is O ( m log n + n 2 ) where m is the number of edges in the disk graph, and it works for disks of arbitrary radii. For graphical clique, our proposed algorithm works for unit disks (i.e., of same radii) and the worst case time complexity is O ( n 2 + m ( n + K 3 ) ) ; m is the number of edges in the unit disk intersection graph and K is the size of the largest clique in that graph. It uses O ( n 3 ) time in-place computation of maximum matching in a bipartite graph, where the vertices are given in an array, and the existence of an edge between a pair of vertices can be checked by an oracle on demand (from problem specification) in O ( 1 ) time. This problem is of independent interest. All these algorithms need O ( 1 ) work space in addition to the input array. [ABSTRACT FROM AUTHOR]

Abstract: A hypergraph is a set family defined on vertex set . The dual of is the set of minimal subsets of such that for any . The computation of the dual is equivalent to several problems, such as minimal hitting set enumeration of a subset family, generation problem for maximal frequent and minimal infrequent sets, and so on. Although many algorithms have been proposed for the problem, to the best of our knowledge, none of them can work on large-scale input with a large number of minimal hitting sets. In this paper, we propose efficient algorithms for checking the minimality and pruning methods. These algorithms enable us to construct time effective and polynomial space dualization algorithms. The computational experiments show that our algorithms are quite fast even for large-scale input for which existing algorithms do not terminate in a practical time. [Copyright &y& Elsevier]