Finding Exact and Maximum Occurrences of Protein Complexes in Protein-Protein Interaction Graphs

Comparing genomic properties of multiple species at varying evolutionary distances is a powerful approach for studying biological and evolutionary principles. In the context of comparative analysis of protein-protein interaction graphs, we use a graph-based formalism to detect the preservation of a given protein complex G in the protein-protein interaction graph H of another species with respect to (w.r.t.) orthologous proteins, i.e., proteins in different species that have diverged in the two lineages from a common ancestor. Two problems are considered: the Exact-(μG, μH)-Matching problem and the Max-(μG, μH)-Matching problem, where μG (resp. μH) denotes in both problems the maximum number of orthologous proteins in H (resp. G) of a protein in G (resp. H). Following [FLV04], the Exact-(μG, 1)-Matching problem asks for an injective homomorphism of G to H w.r.t. orthologous proteins. The optimization version is called the Max-(μG, 1)-Matching problem and is concerned with finding an injective mapping of a graph G to a graph H w.r.t. orthologous proteins that matches as many edges of G as possible. For both problems, the emphasis here is clearly on bounded degree graphs G and H, and extremal small values of parameters μG and μH . We prove that the Exact-(μG, 1)-Matching problem for ∆(G) ≤ 2 is polynomial-time solvable for any constant μG, but that the Exact-(3, 2)-Matching problem is NP-complete for bipartite graphs G and H with ∆(G) ≤ 2 and ∆(H) ≤ 3. Concerning the optimization version, the Max-(2, 1)-Matching problem is proved to be APX-hard for ∆(G) ≤ 3 and ∆(H) ≤ 3. To counterbalance this discouraging result, it is shown here that the Max-(μG, 1)-Matching problem for bounded degree graph G is fixed-parameter-tractable parameterized by the number of matched edges and is approximable within ratio 2 d3∆(G)/5e for even ∆(G) and ratio 2 d(3∆(G) + 2)/5e for odd ∆(G), for any ∆(H) and any constant μG, thereby proving that the Max-(2, 1)-Matching problem for bounded degree graphs G and H is in APX, and hence APX-complete. Still in case μH = 1, we complement these results by giving a fast randomized algorithm that achieves a ratio μG , for any constant μG.