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: 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 ). 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.