The paper presents several results on the exact matching problem and related problems:
It proposes a deterministic FPT algorithm for the exact matching problem, parameterized by the independence number and the minimum size of an odd cycle transversal. This extends previous results for bipartite graphs.
The paper also considers the bounded correct parity matching problem, a relaxed variant of the exact matching problem. It shows that this problem can be solved by a deterministic FPT algorithm parameterized by the minimum size of an odd cycle transversal.
For the correct parity matching problem, a further relaxation of the exact matching problem, the paper shows that a slight generalization of an equivalent problem is NP-hard, even for bipartite graphs with a unique edge of weight 1.
The paper discusses several related problems, such as the odd alternating cycle problem and the disjoint augmenting path problem, and establishes their NP-hardness.
The paper also presents a heuristic approach to speed up the FPT algorithm for the exact matching problem by exploiting the unbalanced bipartization problem.
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