Algorithmic and Complexity Results for Decompositions of Biological Networks into Monotone Subsystems

Abstract

A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson [20]. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model, and it was found to perform close to optimally.

Proximity of intracellular regulatory networks to monotone systems

“…4 illustrates this concept on a toy network model. This algorithm was applied because removing all negative loops is NP-hard, although approximation algorithms for this task have been developed [29].…”

Optimal Edge Deletions for Signed Graph Balancing

“…In particular, motivated by the mathematical analysis of large-scale biological networks, DasGupta et al [3] use it to model the concept of "monotone subsystems", under the name of "sign-consistent graphs". A graph G = (V, E) with edges labeled by h : E → {0, 1} is balanced if there is a vertex coloring f : V → {0, 1} such that ∀{u, v} ∈ E : h({u, v}) ≡ (f (u) + f (v)) (mod 2).…”

“…Therefore, in the following we use the notations "=-edge" and " =-edge" instead. The task of decomposing a network into monotone subsystems is then formulated as the graph modification problem Balanced Subgraph, called Undirected Labeling Problem by DasGupta et al [3]. This problem also finds numerous other applications, e. g., in statistical physics and integrated circuit fabrication techniques [2,22].…”

“…Balanced Subgraph is a generalization of the NPhard Maximum Cut problem in graphs. Based on semidefinite programming for Max2SAT [20], DasGupta et al [3] developed an approximation algorithm that guarantees a solution with at least 87.9% of the number of edges of an optimal solution. They further showed that this approximation factor cannot be improved arbitrarily.…”

“…4). We experimented with the real-world biological instances provided by DasGupta et al [3]. We need similar amounts of time, but can solve them optimally.…”