Etsuji Tomita

The worst-case time complexity for generating all maximal cliques and computational experiments

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron-Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3n/3) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3n/3 maximal cliques in an n-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.

An efficient branch-and-bound algorithm for finding a maximum clique

We present an exact and efficient branch-and-bound algorithm for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular kind of graph. It employs approximate coloring and appropriate sorting of vertices to get an upper bound on the size of a maximum clique. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that our algorithm remarkably outperforms other existing algorithms in general. It has been successfully applied to interesting problems in bioinformatics, image processing, the design of quantum circuits, and the design of DNA and RNA sequences for biomolecular computation.

A simple and faster branch-and-bound algorithm for finding a maximum clique

This paper proposes new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, 95–111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Consequently, it is shown by extensive computational experiments that MCS is remarkably faster than MCR and other existing algorithms. It is faster than the other algorithms by an order of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graphs. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs.

The Worst-Case Time Complexity for Generating All Maximal Cliques

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in Bron and Kerbosch’s algorithm. All maximal cliques generated are output in a tree-like form. Then we prove that its worst-case time complexity is O(3 n/3) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3 n/3 cliques in an n-vertex graph.

A Simple Algorithm for Finding a Maximum Clique and Its Worst-Case Time Complexity

This paper proposes a new algorithm MAXCLIQUE which finds a maximum clique in an undirected graph with n vertices, and shows that its worst-case time complexity is O(2n/2.863). For the dual problem of finding a maximum independent set of vertices, Tarjan et al. already have proposed an algorithm of worst-case time complexity O(2n/3) [2]. However, by comparison, our algorithm is remarkably simpler, and it was confirmed that it runs faster when two algorithms were used for several random graphs and their average running times were measured.

Efficient Algorithms for Finding Maximum and Maximal Cliques: Effective Tools for Bioinformatics

Many problems can be formulated as graphs where a graph consists of a set of vertices and a set of edges, in which the vertices stand for objects in question and the edges stand for some relations among the objects. A clique is a subgraph in which all pairs of vertices are mutually adjacent. Thus, a maximum clique stands for a maximum collection of objects which are mutually related in some specified criterion. The so called maximum clique problem is one of the original 21 problems shown to be NP-complete by R. Karp (19). Therefore, it is strongly believed that the maximum clique problem is not solvable easily, i.e., it is not solvable in polynomial-time. Nevertheless, much work has been done on this problem, experimentally and theoretically. It attracts much attention especially recently since it has found many practical applications to bioinformatics (see, e.g., (2; 15; 27; 28; 37; 3; 9; 4; 8; 14; 55; 23; 25; 22; 13)) and many others (see, e.g., excellent surveys (34; 5), and (17; 20; 31; 49; 54; 51)). This chapter presents efficient algorithms for finding a maximum clique and maximal cliques as effective tools for bioinformatics, and shows our successful applications of these algorithms to bioinformatics.

A clique-based method for the edit distance between unordered trees and its application to analysis of glycan structures.

BackgroundMeasuring similarities between tree structured data is important for analysis of RNA secondary structures, phylogenetic trees, glycan structures, and vascular trees. The edit distance is one of the most widely used measures for comparison of tree structured data. However, it is known that computation of the edit distance for rooted unordered trees is NP-hard. Furthermore, there is almost no available software tool that can compute the exact edit distance for unordered trees.ResultsIn this paper, we present a practical method for computing the edit distance between rooted unordered trees. In this method, the edit distance problem for unordered trees is transformed into the maximum clique problem and then efficient solvers for the maximum clique problem are applied. We applied the proposed method to similar structure search for glycan structures. The result suggests that our proposed method can efficiently compute the edit distance for moderate size unordered trees. It also suggests that the proposed method has the accuracy comparative to those by the edit distance for ordered trees and by an existing method for glycan search.ConclusionsThe proposed method is simple but useful for computation of the edit distance between unordered trees. The object code is available upon request.

A Clique-Based Method Using Dynamic Programming for Computing Edit Distance Between Unordered Trees

Many kinds of tree-structured data, such as RNA secondary structures, have become available due to the progress of techniques in the field of molecular biology. To analyze the tree-structured data, various measures for computing the similarity between them have been developed and applied. Among them, tree edit distance is one of the most widely used measures. However, the tree edit distance problem for unordered trees is NP-hard. Therefore, it is required to develop efficient algorithms for the problem. Recently, a practical method called clique-based algorithm has been proposed, but it is not fast for large trees. This article presents an improved clique-based method for the tree edit distance problem for unordered trees. The improved method is obtained by introducing a dynamic programming scheme and heuristic techniques to the previous clique-based method. To evaluate the efficiency of the improved method, we applied the method to comparison of real tree structured data such as glycan structures. For large tree-structures, the improved method is much faster than the previous method. In particular, for hard instances, the improved method achieved more than 100 times speed-up.

A direct branching algorithm for checking equivalence of some classes of deterministic pushdown automata

A new direct algorithm is presented for checking equivalence of some classes of deterministic pushdown automata (dpdas), after Korenjak and Hopcrofts branching algorithm. It is not only powerful enough to be applicable to two dpdas accepting by empty stack, one of which is real-time, but also simple even for dpdas in lower subclasses. This is the first time the branching algorithm has been used to give such a general decision procedure without ever “mixing” the two languages in question. In other words, it deals with only the equivalence equation whose left-hand side consists of a pure reachable configuration of one dpda and whose right-hand side that of the other.

Original Contribution: Separability of internal representations in multilayer perceptrons with application to learning

It is mathematically investigated as to what kind of internal representations are separable by single output units of three-layer perceptrons. A topological description is given for the necessary and sufficient condition that hidden layer representations of input patterns are separable by the output unit. An efficient algorithm is proposed for checking whether or not a hidden layer representation is linearly separable and, if not, for specifying inseparable portions in the partition. Application of the algorithm to learning of three-layer perceptrons is presented in which redundant units are utilized to reduce inseparable partition into separable one. Polynomial learnability from examples and queries is shown for the proposed learning algorithm.