Fully derandomize the isolation lemma for linear matroid intersection in NC

Establish a deterministic NC procedure that, given two linear matroids M1=(S, I1) and M2=(S, I2) over the same ground set, constructs polynomially bounded integer weights w:S→Z such that the minimum-weight maximum-size common independent set in I1∩I2 is unique, thereby fully derandomizing the isolation lemma for linear matroid intersection within the NC hierarchy.

Background

The isolation lemma is a central randomized tool: assigning random weights to elements of the ground set isolates a unique optimal solution with high probability. For linear matroid intersection, Narayanan, Saran, and Vazirani used this lemma to design RNC algorithms, and Gurjar–Thierauf obtained quasi-NC derandomization with quasi-polynomially bounded weights.

This paper provides a CLP derandomization for linear matroid intersection but does not resolve the stronger goal of achieving an NC derandomization. Thus, the longstanding challenge is to obtain a deterministic NC method that guarantees isolation via polynomially bounded weights.