Deterministic polynomial-time algorithm for Exact Matching in general graphs

Determine whether there exists a deterministic polynomial-time algorithm for the Exact Matching problem in general graphs; specifically, given a 0/1-weighted graph G and an integer k, decide whether G has a perfect matching of weight exactly k.

Background

Beyond the specific derandomization of MVV, the broader question is whether EM admits any deterministic polynomial-time algorithm in general graphs. The paper notes that several special cases (e.g., complete or complete bipartite graphs, K3,3-minor-free graphs, bounded-genus graphs) have deterministic solutions, but the general case remains unresolved.

This problem sits at the core of understanding whether the parity-constrained matching structures that MVV handle via randomness can be captured deterministically across all graphs.