Parameterized Complexity of Conflict-Free Matchings and Paths

Abstract

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 Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching 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-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid.

Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

“…There is a long history of study on treewidth and parameterization on treewidth [3,4,6]. For a wide range of hard problems, algorithms with running time of the form f (k) • n O (1) exist if the input graph comes with a tree decomposition of width k. Moreover, in many cases, we have good understanding of the best possible form of f (k) in the running time (under suitable complexity assumptions, such as the Exponential Time Hypothesis (ETH) or the Strong Exponential Time Hypothesis (SETH) put forward by Impagliazzo and Paturi [25]). Marx et al [32] studied the B-Factor problem for this viewpoint, parameterized by treewidth for finite, fixed lists B.…”

“…They showed that a combination of standard dynamic programming techniques with fast subset convolution (cf. [38]) give a (max B + 1) tw n O (1) time algorithm for the decision, minimization, maximization, and counting versions. Furthermore, this form of running time is essentially optimal conditioned on the SETH.…”

“…We can count in time (max B + 1) tw n O (1) the solutions of a certain size for a B-Factor instance if we are given a tree decomposition of width tw. For any > 0, there is no (max B + 1 − ) pw n O (1) algorithm for the following problems, even if we are given a path decomposition of width pw, unless SETH (resp. #SETH) fails:…”

“…. , max B at each vertex, which is the main reason we need (max B + 1) tw n O (1) running time. In the X-AntiFactor problem, a vertex can have degree 0, 1, .…”

“…, max X, or more than max X in a partial solution. Therefore, the natural running time we expect is (max X + 2) tw n O (1) . We show that this running time can be achieved, but requires some modification the way the convolution is done to handle the state "degree more than max X."…”