FPT–matrix-multiplication time for parity k-matchings (⊕Match)

Determine whether the parameterized problem ⊕Match—computing the parity of the number of k-matchings in an uncoloured graph on n vertices—admits an algorithm running in time f(k)·O(n^ω), where ω is the matrix multiplication exponent and f is a computable function. This asks for an explicit fixed-parameter tractable algorithm with matrix-multiplication exponent for ⊕Match, or evidence ruling out such an improvement under standard assumptions.

Background

The paper proves that counting colourful k-matchings modulo 2 (⊕ColMatch) can be solved in FPT–matrix-multiplication time, and shows that neither ⊕ColMatch nor ⊕Match can be solved in FPT–near-linear time unless the Triangle Conjecture fails. However, their uncoloured holant classification does not extend to modular counting, leaving the precise FPT exponent for ⊕Match unresolved.

Establishing an FPT–matrix-multiplication time algorithm for ⊕Match would align the parity version in uncoloured graphs with its colourful counterpart and sharpen the fine-grained landscape of modular counting problems parameterized by k. Conversely, a conditional lower bound would separate the colourful and uncoloured settings for parity matchings.