Parameterized Complexity of Disconnected Matchings

The quest to match entities within graphs is a cornerstone of graph theory, and such algorithmic approaches have been investigated for centuries. Traditionally, given a graph G, the problem is to find a maximum size matching in G. Subsequently, the goal has been to find a matching M such that the graph induced on the endpoints of the edges of M, G[V_M], has some additional property P, such as induced matching, acyclicity, connectivity, disconnectedness, etc. In this paper, we focus on the property of disconnectedness. In particular, we consider the following problem defined by Gomes et al. [TCS ’23]: given a graph G, and two positive integers k and c; we want to know if there exists a matching M of size at least k such that G[V_M] has at least c connected components? We call this the Disconnected Matching problem. We show the following results. When c is a constant, the problem admits an FPT algorithm with respect to the solution size k. Our algorithm runs in time 9^ckn^O(c), where n denotes the number of vertices in G. Moreover, we show that we cannot hope for a polynomial size kernel with respect to c+k unless NP⊆coNP/poly. For arbitrary c, we cannot hope for an FPT algorithm with respect to the solution size due to the W[1]-hardness of Induced Matching.We present an algorithm that runs in time O^⋆(c·3^ℓ) (O^⋆ hides the polynomial factor in the running time), where ℓ is the size of a minimum vertex cover. This is in contrast to the algorithm proposed by Chaudhary and Zehavi [WG ’23] that runs in O^⋆((3c)^tw) time, where tw is the treewidth of G and so must depend on c.We also design an exact algorithm that runs in O^⋆((2-ϵ)ⁿ) time for some fixed 0<ϵ<1, where n is the number of vertices in G. When c is a constant, the problem admits an FPT algorithm with respect to the solution size k. Our algorithm runs in time 9^ckn^O(c), where n denotes the number of vertices in G. Moreover, we show that we cannot hope for a polynomial size kernel with respect to c+k unless NP⊆coNP/poly. For arbitrary c, we cannot hope for an FPT algorithm with respect to the solution size due to the W[1]-hardness of Induced Matching. We present an algorithm that runs in time O^⋆(c·3^ℓ) (O^⋆ hides the polynomial factor in the running time), where ℓ is the size of a minimum vertex cover. This is in contrast to the algorithm proposed by Chaudhary and Zehavi [WG ’23] that runs in O^⋆((3c)^tw) time, where tw is the treewidth of G and so must depend on c. We also design an exact algorithm that runs in O^⋆((2-ϵ)ⁿ) time for some fixed 0<ϵ<1, where n is the number of vertices in G.