Abstract:
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 has 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 TR₁ with no critical edges, is known to be MAX-SNP-hard, admits a polynomial time algorithm with an approximation ratio of 1.617+\eps for any constant \eps>0 and can be solved in linear time for directed acyclic graphs.

(2) A 2-approximation algorithm exists for TR₁.

In this paper, our contributions are as follows:

(a) We observe that TR_p, for any integer p>0, can be solved in linear time for directed acyclic graphs.

(b) We provide a 1.78-approximation for TR₁ that improves the 2-approximation mentioned in (2) above.

(c) We provide a 2+o(1)-approximation for TR_p on general graphs for any prime p>0.