An algorithm for weighted fractional matroid matching

LetM be a matroid on ground set E. A subset l ⊆ E is called a line when r(l) ∈ {1, 2}. Given a set of lines L = {l1, . . . , lk} in M , a vector x ∈ RL+ is called a fractional matching when ∑ l∈L xla(F )l ≤ r(F ) for every flat F ofM . Here a(F )l is equal to 0 when l∩F = ∅, equal to 2 when l ⊆ F and equal to 1 otherwise. We refer to ∑ l∈L xl as the size of x. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give an efficient algorithm to find for given weight function w : L → Q a maximum weight fractional matching.