Inferring (Biological) Signal Transduction Networks via Transitive Reductions of Directed Graphs

In this paper we consider the p-ary transitive reduction (TRp) 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 TRp has been investigated before in dierent contexts; the best previous results are as follows: (1) The minimum equivalent digraph problem, that correspond to a special case of TR1 with no critical edges, is known to be MAX-SNP-hard, admits a polynomial time algorithm with an approximation ratio of 1:617+ " for any constant "> 0 [13] and can be solved in linear time for directed acyclic graphs [1]. (2) A 2-approximation algorithm exists for TR1 [9, 15]. In this paper, our contributions are as follows: We observe that TRp, for any integer p> 0, can be solved in linear time for directed acyclic graphs using the ideas in [1]. We provide a 1:78-approximation for TR1 that improves the 2-approximation mentioned in (2) above. We provide a 2 + o(1)-approximation for TRp on general graphs for any prime p> 0