Finding and Counting Vertex-Colored Subtrees

Finding maximum colorful subtrees in practice

In metabolomics and other fields dealing with small compounds, mass spectrometry is applied as a sensitive high-throughput technique. Recently, fragmentation trees have been proposed to automatically analyze the fragmentation mass spectra recorded by such instruments. Computationally, this leads to the problem of finding a maximum weight subtree in an edge-weighted and vertex-colored graph, such that every color appears, at most once in the solution. We introduce new heuristics and an exact algorithm for this Maximum Colorful Subtree problem and evaluate them against existing algorithms on real-world and artificial datasets. Our tree completion heuristic consistently scores better than other heuristics, while the integer programming-based algorithm produces optimal trees with modest running times. Our fast and accurate heuristic can help determine molecular formulas based on fragmentation trees. On the other hand, optimal trees from the integer linear program are useful if structure is relevant, for example for tree alignments.

RANGI: a fast list-colored graph motif finding algorithm

Given a multiset of colors as the query and a list-colored graph, i.e., an undirected graph with a set of colors assigned to each of its vertices, in the NP-hard list-colored graph motif problem the goal is to find the largest connected subgraph such that one can select a color from the set of colors assigned to each of its vertices to obtain a subset of the query. This problem was introduced to find functional motifs in biological networks. We present a branch-and-bound algorithm named RANGI for finding and enumerating list-colored graph motifs. As our experimental results show, RANGI's pruning methods and heuristics make it quite fast in practice compared to the algorithms presented in the literature. We also present a parallel version of RANGI that achieves acceptable scalability.

Parameterized algorithmics for finding connected motifs in biological networks

We study the NP-hard LIST-COLORED GRAPH MOTIF problem which, given an undirected list-colored graph G = (V, E) and a multiset M of colors, asks for maximum-cardinality sets S ⊆ V and M' ⊆ M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M'. LIST-COLORED GRAPH MOTIF has applications in the analysis of biological networks. We study LIST-COLORED GRAPH MOTIF with respect to three different parameterizations. For the parameters motif size |M| and solution size |S|, we present fixed-parameter algorithms, whereas for the parameter |V| - |M|, we show W[1]-hardness for general instances and achieve fixed-parameter tractability for a special case of LIST-COLORED GRAPH MOTIF. We implemented the fixed-parameter algorithms for parameters |M| and |S|, developed further speed-up heuristics for these algorithms, and applied them in the context of querying protein-interaction networks, demonstrating their usefulness for realistic instances. Furthermore, we show that extending the request for motif connectedness to stronger demands, such as biconnectedness or bridge-connectedness leads to W[1]-hard problems when the parameter is the motif size |M|.