A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision

Given two matroids on the same ground set, the matroid intersection problem asks for a common base, i.e., a subset of the ground set that is a base in both the matroids. The weighted version of the problem asks for a common base with maximum weight. In the case of linearly representable matroids, the weighted version is known to have a randomized parallel (RNC) algorithm based on the isolation lemma, when the given weights are polynomially bounded (Narayanan et al. in SIAM J Comput 23(2): 387–397, 1994). Finding a deterministic parallel (NC) algorithm, even for the unweighted decision question, has been a long-standing open question. The above RNC algorithm can be viewed as a randomized reduction from weighted search to weighted decision, which works for arbitrary matroids. We derandomize this reduction, i.e., we give an NC algorithm for weighted matroid intersection search using oracle access to its decision version.