Approximation Hardness of the Cross-Species Conserved Active Modules Detection Problem

Biological network comparison is an essential but algorithmically challenging approach for the analysis of underlying data. A typical example is looking for certain subgraphs in a given network, such as subgraphs that maximize some function of their nodes’ weights. However, the corresponding maximum-weight connected subgraph (mwcs) problem is known to be hard to approximate. In this contribution, we consider the problem of the simultaneous discovery of maximum weight subgraphs in two networks, whose nodes are matched by a mapping: the maximum-weight cross-connected subgraphs (mwccs) problem. We provide inapproximability results for this problem. These results indicate that the complexity of the problem is conditioned both by the nature of the mapping function and by the topologies of the two networks. In particular, we show that the problem is inapproximable even when the mapping is an injective function and the input graphs are two binary trees. We also prove that it remains hard to approximate when the mapping is a bijective function and the input graphs are a graph and a binary tree. We further analyze a variant of the mwcs problem where the networks’ nodes are assigned both a weight and a contribution value, that we call maximum-weight ratio-bounded connected subgraph (mwrbcs). We provide a polynomial time algorithm for bounded-degree trees and an efficient dynamic programming solution for cycles. These algorithms allow us to derive a polynomial solution for mwccs applicable when (i) mwrbcs is polynomially solvable for one of the graphs and (ii) the set of connected induced subgraphs of the other graph is polynomially enumerable.