Deterministic polynomial-time algorithm for Bounded Correct Parity Matching in general graphs

Determine whether the Bounded Correct Parity Matching problem can be solved by a deterministic polynomial-time algorithm in general graphs; namely, given a 0/1-weighted graph G and an integer k, decide whether G has a perfect matching of weight k′ such that k′ < k and k′ ≡ k (mod 2).

Background

El Maalouly, Steiner, and Wulf provided a deterministic polynomial-time algorithm for BCPM in bipartite graphs using dynamic programming on an equivalent formulation. The present paper observes that extending this approach to general graphs appears difficult due to blossoms—structures that complicate alternating-path-based methods in non-bipartite graphs.

Despite FPT results and reductions tying EM to BCPM under certain parameters, whether BCPM itself admits a deterministic polynomial-time algorithm for general graphs explicitly remains open.

References

Their algorithm is based on a standard dynamic programming approach for an equivalent problem. It seems difficult to be generalized to general graphs (due to existence of so-called blossoms), and it remains open whether BCPM can be deterministically solved in polynomial time or not.

— An FPT Algorithm for the Exact Matching Problem and NP-hardness of Related Problems (2405.02829 - Murakami et al., 5 May 2024) in Section 1.1 (FPT Algorithms for Exact Matching Problem)