Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M (n)) time delay and in O(n 2 ) space and the other runs with O(∆ 4 ) time delay and in O(n + m) space, where ∆ denotes the maximum degree of G, M (n) denotes the time needed to multiply two n × n matrices, and the latter one requires O(nm) time as a preprocessing. For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques. The first algorithm runs with O(M (n)) time delay and in O(n 2 ) space, which immediately follows from the algorithm for the nonbipartite case. The second one runs with O(∆ 3 ) time delay and in O(n + m) space, and the last one runs with O(∆ 2 ) time delay and in O(n + m + N ∆) space, where N denotes the number of all maximal bipartite cliques in G and both algorithms require O(nm) time as a preprocessing. Our algorithms improve upon all the existing algorithms, when G is either dense or sparse. Furthermore, computational experiments show that our algorithms for sparse graphs have significantly good performance for graphs which are generated randomly and appear in real-world problems.

Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose precise tractability status (in terms of polynomial-time solvability) is still unknown, and now open for more than 25 years. In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan (J. Algorithms, 21:618-628, 1996), which showed that the problem is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. We consider computational aspects including limited nondeterminism, probabilistic computation, parallel and learning-based algorithms, and implementations and experimental results from the literature. The paper closes with open issues for further research.

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Abstract. Let A be an m×n binary matrix, t ∈ {1, . . . , m} be a threshold, and ε > 0 be a positive parameter. We show that given a family of O(n ε ) maximal t-frequent column sets for A, it is NP-complete to decide whether A has any further maximal t-frequent sets, or not, even when the number of such additional maximal t-frequent column sets may be exponentially large. In contrast, all minimal t-infrequent sets of columns of A can be enumerated in incremental quasi-polynomial time. The proof of the latter result follows from the inequality α ≤ (m − t + 1)β, where α and β are respectively the numbers of all maximal t-frequent and all minimal t-infrequent sets of columns of the matrix A. We also discuss the complexity of generating all closed t-frequent column sets for a given binary matrix.

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In this paper, we consider a sink location in a dynamic network which consists of a graph with capacities and transit times on its arcs. Given a dynamic network with initial supplies at vertices, the problem is to find a vertex v as a sink in the network such that we can send all the initial supplies to v as quickly as possible.We present an O(n log 2 n) time algorithm for the sink location problem in a dynamic network of tree structure, where n is the number of vertices in the network. This improves upon the existing O(n 2 )-time bound. As a corollary, we also show that the quickest transshipment problem can be solved in O(n log 2 n) time if a given network is a tree and has a single sink. Our results are based on data structures for representing tables (i.e., sets of intervals with their height), which may be of independent interest.

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(χ(n) · log n) suitably guessed bits, where χ(n) is given by χ(n) χ(n) = n; note that χ(n) = o(log n). This result sheds new light on the complexity of this important problem.

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary "true n-vectors" (or "positive examples") and a set of "false n-vectors" (or "negative examples"), we establish a Boolean function (or an extension) f , so that f is true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions and so on. The class of functions represents our a priori knowledge or hypothesis about the extension f , which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extension f having the minimum number of misclassifications. Both problems, i.e. the problem of finding an extension within a specified class of Boolean functions, and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.