Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications

The problem of finding a maximum clique or enumerating all maximal cliques is very important and has been explored in several excellent survey papers. Here, we focus our attention on the step-by-step examination of a series of branch-and-bound depth-first search algorithms: Basics, MCQ, MCR, MCS, and MCT. Subsequently, as with the depth-first search as above, we present our algorithm, CLIQUES, for enumerating all maximal cliques. Finally, we describe some of the applications of the algorithms and their variants in bioinformatics, data mining, and other fields

An extended direct branching algorithm for checking equivalence of deterministic pushdown automata

AbstractThis paper extends the direct branching algorithm of [25] for checking equivalence of deterministic pushdown automata. It does so by providing a technique called ‘halting’ for dealing with nodes with unbounded degree in the comparison tree. This may occur when a skipping step may be applied infinitely many times to a certain node, as a result of infinite sequences of ε-moves.This extension allows the algorithm to check equivalence of two deterministic pushdown automata when none of them is real-time, but in a certain condition that properly contains a case where one of them is real-time strict

Another Time-Complexity Analysis for Maximal Clique Enumeration Algorithm CLIQUES

We revisit the maximal clique enumeration algorithm CLIQUES that appeared in Theoretical Computer Science 2006. It is proved to work in O(3n/3)-time in the worst-case for an n vertex graph. In this note, we extend the time-complexity analysis with respect to the number of maximal cliques, an issue that was left as an open problem since TCS 2006

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

[Background]Measuring 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. [Results]In 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. [Conclusions]The proposed method is simple but useful for computation of the edit distance between unordered trees. The object code is available upon request

A Much Faster Algorithm for Finding a Maximum Clique

We present improvements to a branch-and-bound maximumclique-finding algorithm MCS (WALCOM 2010, LNCS 5942, pp. 191–203) that was shown to be fast. First, we employ an efficient approximation algorithm for finding a maximum clique. Second, we make use of appropriate sorting of vertices only near the root of the search tree. Third, we employ a lightened approximate coloring mainly near the leaves of the search tree. A new algorithm obtained from MCS with the above improvements is named MCT. It is shown that MCT is much faster than MCS by extensive computational experiments. In particular, MCT is shown to be faster than MCS for gen400 p0.9 75 and gen400 p0.9 65 by over 328,000 and 77,000 times, respectively

Clique-based data mining for related genes in a biomedical database

<p>Abstract</p> <p>Background</p> <p>Progress in the life sciences cannot be made without integrating biomedical knowledge on numerous genes in order to help formulate hypotheses on the genetic mechanisms behind various biological phenomena, including diseases. There is thus a strong need for a way to automatically and comprehensively search from biomedical databases for related genes, such as genes in the same families and genes encoding components of the same pathways. Here we address the extraction of related genes by searching for densely-connected subgraphs, which are modeled as cliques, in a biomedical relational graph.</p> <p>Results</p> <p>We constructed a graph whose nodes were gene or disease pages, and edges were the hyperlink connections between those pages in the Online Mendelian Inheritance in Man (OMIM) database. We obtained over 20,000 sets of related genes (called 'gene modules') by enumerating cliques computationally. The modules included genes in the same family, genes for proteins that form a complex, and genes for components of the same signaling pathway. The results of experiments using 'metabolic syndrome'-related gene modules show that the gene modules can be used to get a coherent holistic picture helpful for interpreting relations among genes.</p> <p>Conclusion</p> <p>We presented a data mining approach extracting related genes by enumerating cliques. The extracted gene sets provide a holistic picture useful for comprehending complex disease mechanisms.</p