Heuristic algorithms for the Maximum Colorful Subtree problem

TL;DR

This paper evaluates heuristic algorithms for the NP-hard Maximum Colorful Subtree problem in metabolomics, finding one heuristic that efficiently ranks solutions highly and speeds up computations significantly.

Contribution

It introduces an intermediate evaluation method for heuristics in the problem and identifies a heuristic that effectively ranks correct solutions and accelerates processing.

Findings

One heuristic consistently ranks the correct solution in a top position.

The best heuristic solutions have scores close to the optimal.

Solution structures can differ significantly from the optimal despite high scores.

Abstract

In metabolomics, small molecules are structurally elucidated using tandem mass spectrometry (MS/MS); this resulted in the computational Maximum Colorful Subtree problem, which is NP-hard. Unfortunately, data from a single metabolite requires us to solve hundreds or thousands of instances of this problem; and in a single Liquid Chromatography MS/MS run, hundreds or thousands of metabolites are measured. Here, we comprehensively evaluate the performance of several heuristic algorithms for the problem against an exact algorithm. We put particular emphasis on whether a heuristic is able to rank candidates such that the correct solution is ranked highly. We propose this "intermediate" evaluation because evaluating the approximating quality of heuristics is misleading: Even a slightly suboptimal solution can be structurally very different from the true solution. On the other hand, we cannot structurally evaluate against the ground truth, as this is unknown. We find that one particular heuristic consistently ranks the correct solution in a top position, allowing us to speed up computations about 100-fold. We also find that scores of the best heuristic solutions are very close to the optimal score; in contrast, the structure of the solutions can deviate significantly from the optimal structures.