Parameterized Complexity of Conflict-Free Matchings and Paths - Agrawal, Akanksha; Jain, Pallavi; Kanesh, Lawqueen; Saurabh, Saket (Algorithmica, 2020)
An input to a conflict-free variant of a classical problem Γ , called Conflict-Free Γ , consists of an instance I of Γ coupled with a graph H , called the conflict graph . A solution to Conflict-Free Γ 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.