Complexity of modification problems for reciprocal best match graphs

Abstract

Reciprocal best match graphs (RBMGs) are vertex colored graphs whose vertices represent genes and the colors the species where the genes reside. Edges identify pairs of genes that are most closely related with respect to an underlying evolutionary tree. In practical applications this tree is unknown and the edges of the RBMGs are inferred by quantifying sequence similarity. Due to noise in the data, these empirically determined graphs in general violate the condition of being a "biologically feasible" RBMG. Therefore, it is of practical interest in computational biology to correct the initial estimate. Here we consider deletion (remove at most k edges) and editing (add or delete at most k edges) problems. We show that the decision version of the deletion and editing problem to obtain RBMGs from vertex colored graphs is NP-hard. Using known results for the so-called bicluster editing, we show that the RBMG editing problem for 2-colored graphs is fixedparameter tractable.A restricted class of RBMGs appears in the context of orthology detection. These are cographs with a specific type of vertex coloring known as hierarchical coloring. We show that the decision problem of modifying a vertex-colored graph (either by edge-deletion or editing) into an RBMG with cograph structure or, equivalently, to an hierarchically colored cograph is NP-complete.

From pairs of most similar sequences to phylogenetic best matches

“…Fortunately, these appear to be rare. We expect that methods for orthology assessment can be improved in both reliability and computational performance by combining the accurate estimation of best matches described here with a better understanding of (reciprocal) best match graphs [38,39,60] and their connection with the orthology relation [40]. Since tree-free methods for orthology detection rely on (pairwise) best hits as proxy for reciprocal best matches, we expect that the accuracy of most tools would improve if best matches are supplied as input data.…”

Complete Characterization of Incorrect Orthology Assignments in Best Match Graphs

“…( 2020 ) and the analogous RBMG editing problem Hellmuth et al. ( 2020b ) are NP-hard. Efficient, accurate heuristics are a topic of ongoing research.…”

Complexity of modification problems for best match graphs

“…1 for the case n = m = 3. We emphasize that, in contrast to the definition used in [15], single vertex graphs are not considered as bi-clique.…”

“…The orthology graph of a gene family (with the genes as vertices and undirected edges between orthologous genes) can be shown to be a subgraph of the reciprocal best match graph (RBMG), i.e., the symmetric part of the BMG [11]. This has sparked interest in a characterization of RBMGs [12] and the corresponding graph editing problems [15]. The deletion and the editing problems of 2-colored RBMGs are equivalent to Bicluster Deletion and Bicluster Editing, respectively, a fact that was used to demonstrate NP-hardness for the general, -colored case.…”