Inferring (biological) signal transduction networks via transitive reductions of directed graphs

In this paper we consider the p-ary transitive reduction (TR _p ) problem where p>0 is an integer; for p=2 this problem arises in inferring a sparsest possible (biological) signal transduction network consistent with a set of experimental observations with a goal to minimize false positive inferences even if risking false negatives. Special cases of TR _p have been investigated before in different contexts; the best previous results are as follows: (1) The minimum equivalent digraph problem, that correspond to a special case of TR ₁ with no critical edges, is known to be MAX-SNP-hard, admits a polynomial time algorithm with an approximation ratio of 1.617+ε for any constant ε>0 (Chiu and Liu in Sci. Sin. 4:1396-1400, 1965) and can be solved in linear time for directed acyclic graphs (Aho et al. in SIAM J. Comput. 1(2):131-137, 1972). (2) A 2-approximation algorithm exists for TR ₁ (Frederickson and JàJà in SIAM J. Comput. 10(2):270-283, 1981; Khuller et al. in 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 937-938, 1999). In this paper, our contributions are as follows: • We observe that TR _p , for any integer p>0, can be solved in linear time for directed acyclic graphs using the ideas in Aho et al. (SIAM J. Comput. 1(2):131-137, 1972). • We provide a 1.78-approximation for TR ₁ that improves the 2-approximation mentioned in (2) above. • We provide a 2+o(1)-approximation for TR _p on general graphs for any fixed prime p>1.