Parameterized Complexity of Conflict-Free Matchings and Paths

An input to a conflict-free variant of a classical problem Γ , called Conflict-FreeΓ , consists of an instance I of Γ coupled with a graph H, called the conflict graph. A solution to Conflict-FreeΓ in (I, H) is a solution to I in Γ , which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Maximum Matching (CF-MM) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-MM and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-MM holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-MM when the conflict graph is chordal. Also, we give FPT algorithms for both CF-MM and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-MM and CF-SP, where the conflicting conditions are given by a (representable) matroid.