Mining minimal motif pair sets maximally covering interactions in a protein-protein interaction network

Correlated motif covering (CMC) is the problem of finding a set of motif pairs, i.e., pairs of patterns, in the sequences of proteins from a protein-protein interaction network which describe the interactions in the network as concisely as possible. In other words, a perfect solution for CMC would be a minimal set of motif pairs which describes the interaction behavior perfectly in the sense that two proteins from the network interact if and only if their sequences match a motif pair in the minimal set. In this paper, we introduce and formally define CMC and show that it is closely related to the Red-Blue Set Cover (RBSC) problem and its weighted version (WRBSC) — both well-known NP-hard problems for which there exist several algorithms with known approximation factor guarantees. We prove the hardness of approximation of CMC by providing an approximation factor preserving reduction from RBSC to CMC. We show the existence of a theoretical approximation algorithm for CMC by providing an approximation factor preserving reduction from CMC to WRBSC. We adapt the latter algorithm into a functional heuristic for CMC, called CMC-approx, and experimentally assess its performance and biological relevance.