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Finding Approximate and Constrained Motifs in Graphs
"... Abstract. One of the emerging topics in the analysis of biological networks is the inference of motifs inside a network. In the context of metabolic network analysis, a recent approach introduced in [14], represents the network as a vertexcolored graph, while a motif M is represented as a multiset ..."
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Abstract. One of the emerging topics in the analysis of biological networks is the inference of motifs inside a network. In the context of metabolic network analysis, a recent approach introduced in [14], represents the network as a vertexcolored graph, while a motif M is represented as a multiset of colors. An occurrence of a motif M in a vertexcolored graph G is a connected induced subgraph of G whose vertex set is colored exactly as M. We investigate three different variants of the initial problem. The first two variants, MinAdd and MinSubstitute, deal with approximate occurrences of a motif in the graph, while the third variant, Constrained Graph Motif (or CGM for short), constrains the motif to contain a given set of vertices. We investigate the classical and parameterized complexity of the three problems. We show that MinAdd and MinSubstitute are NPhard, even when M is a set, and the graph is a tree of degree bounded by 4 in which each color appears at most twice. Moreover, we show that MinSubstitute is in FPT when parameterized by the size of M. Finally, we consider the parameterized complexity of the CGM problem, and we give a fixedparameter algorithm for graphs of bounded treewidth, while we show that the problem is W [2]hard, even if the input graph has diameter 2. 1
Binary jumbled pattern matching on trees and treelike structures
 In Proc. of the 21st Annual European Symposium on Algorithm (ESA 2013
, 2013
"... Abstract. Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of length i and has exactly j 1bits. This problem naturally generalizes to vertexlabeled trees and graphs by replacing “substring ” with “connect ..."
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Abstract. Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of length i and has exactly j 1bits. This problem naturally generalizes to vertexlabeled trees and graphs by replacing “substring ” with “connected subgraph”. In this paper, we give an O(n2 / log2 n)time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an O(g2/3n4/3/(logn)4/3)time solution for strings that are compressed by a grammar of size g. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixedparameter tractable with respect to the treewidth w of the graph, even for a constant number of different vertexlabels, thus improving the previous best nO(w) algorithm [ICALP’07]. 1
The challenges of unbounded treewidth in parameterised subgraph counting problems, arXiv:1402.5857 [cs.CC
, 2014
"... Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of the minimal subgraphs having the desired property. In this paper, we give a number of hardness results – for decision, appr ..."
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Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of the minimal subgraphs having the desired property. In this paper, we give a number of hardness results – for decision, approximate counting and exact counting – in the case that this condition on the minimal subgraphs having the desired property does not hold. These results demonstrate that in some cases the bounded treewidth condition is necessary for the existence of an efficient algorithm, and lead to two dichotomies for problems which involve finding or counting multicolour subgraphs. 1
Partial Information Network Queries
, 2013
"... We study the Partial Information Network Query (PINQ) problem, which generalizes two problems that often arise in bioinformatics: the Alignment Network Query (ANQ) problem and the TopologyFree Network Query (TFNQ) problem. In both ANQ and TFNQ we have a pattern P and a graph H, and we seek a subg ..."
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We study the Partial Information Network Query (PINQ) problem, which generalizes two problems that often arise in bioinformatics: the Alignment Network Query (ANQ) problem and the TopologyFree Network Query (TFNQ) problem. In both ANQ and TFNQ we have a pattern P and a graph H, and we seek a subgraph of H that resembles P. ANQ requires knowing the topology of P, while TFNQ ignores it. PINQ fits the scenario where partial information is available on the topology of P. Our main result is a parameterized algorithm that handles inputs for PINQ in which P is a set of trees. This algorithm significantly improves the best known O ∗ running time in solving TFNQ. We also improve the best known O ∗ running times in solving two special cases of ANQ.
Deterministic Parameterized Algorithms for the Graph Motif Problem
, 2014
"... We study the classic Graph Motif problem: given a graph G = (V,E) with a set of colors for each node, and a multiset M of colors, we seek a subtree T ⊆ G, and a coloring of the nodes in T, such that T carries exactly (also with respect to multiplicity) the colors in M. Graph Motif plays a central ro ..."
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We study the classic Graph Motif problem: given a graph G = (V,E) with a set of colors for each node, and a multiset M of colors, we seek a subtree T ⊆ G, and a coloring of the nodes in T, such that T carries exactly (also with respect to multiplicity) the colors in M. Graph Motif plays a central role in the study of pattern matching problems, primarily motivated from the analysis of complex biological networks. Previous algorithms for Graph Motif and its variants either rely on techniques for developing randomized algorithms that, if derandomized, render them inefficient, or the algebraic narrow sieves technique for which there is no known derandomization. In this paper, we present fast deterministic parameterized algorithms for Graph Motif and its variants. Specifically, we give such an algorithm for the more general Graph Motif with Deletions problem, followed by faster algorithms for Graph Motif and other wellstudied special cases. Our algorithms make nontrivial use of representative families, and a novel tool that we call guiding trees, together enabling the efficient construction of the output tree.
Path Sampling: A Fast and Provable Method for Estimating 4Vertex Subgraph Counts∗
"... Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4vertex patterns is highly challenging, and th ..."
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Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. Indeed, even a highly tuned enumeration code takes more than a day on a graph with millions of edges. Most previous work that runs for truly massive graphs employ clusters and massive parallelization. We provide a sampling algorithm that provably and accurately approximates the frequencies of all 4vertex pattern subgraphs. Our algorithm is based on a novel technique of 3path sampling and a special pruning scheme to decrease
Mining integrated semantic networks for drug repositioning opportunities
"... Current research and development approaches to drug discovery have become less fruitful and more costly. One alternative paradigm is that of drug repositioning. Many marketed examples of repositioned drugs have been identified through serendipitous or rational observations, highlighting the need for ..."
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Current research and development approaches to drug discovery have become less fruitful and more costly. One alternative paradigm is that of drug repositioning. Many marketed examples of repositioned drugs have been identified through serendipitous or rational observations, highlighting the need for more systematic methodologies to tackle the problem. Systems level approaches have the potential to enable the development of novel methods to understand the action of therapeutic compounds, but requires an integrative approach to biological data. Integrated networks can facilitate systems level analyses by combining multiple sources of evidence to provide a rich description of drugs, their targets and their interactions. Classically, such networks can be mined manually where a skilled person is able to identify portions of the graph (semantic subgraphs) that are indicative of relationships between drugs and highlight possible repositioning opportunities. However, this approach is not scalable. Automated approaches are required to systematically mine integrated networks for these subgraphs and bring them to the