Abstract. A graph G is called a block graph if each maximal 2-connected component of G is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph G and a positive integer k and the objective to check whether there exists a subset S ⊆ V (G) of size at most k such that the graph induced on V (G) \ S is a block graph. In this paper we give an FPT algorithm with running time 4 k n O(1) and a polynomial kernel of size O(k 4 ) for Block Graph Vertex Deletion. The running time of our FPT algorithm improves over the previous best algorithm for the problem that ran in time 10 k n O(1) and the size of our kernel reduces over the previously known kernel of size O(k 9 ). Our results are based on a novel connection between Block Graph Vertex Deletion and the classical Feedback Vertex Set problem in graphs without induced C4 and K4 − e. To achieve our results we also obtain an algorithm for Weighted Feedback Vertex Set running in time 3.618 k n O(1) and improving over the running time of previously known algorithm with running time 5 k n O(1) .
We study exact algorithms for EUCLIDEAN TSP in R d . In the early 1990s algorithms with n O( √ n)running time were presented for the planar case, and some years later an algorithm with n O(n 1−1/d ) running time was presented for any d 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on EUCLIDEAN TSP, except for a lower bound stating that the problem admits no 2 O(n 1−1/d−ε ) algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of EUCLIDEAN TSP by giving a 2 O(n 1−1/d ) algorithm and by showing that a 2 o(n 1−1/d ) algorithm does not exist unless ETH fails. affirmatively by Arora [1] who provided a PTAS with running time n(log n) O( √ d/ε) d−1 . Independently, Mitchell [18] designed a PTAS in R 2 . The running time was improved to 2 (1/ε) O(d) n + (1/ε) O(d) n log n by Rao and Smith [22]. Hence, the computational complexity of the approximation problem has essentially been settled. Results on exact algorithms for EUCLIDEAN TSP-these are the topic of our paper-are also quite different from those on the general problem. The best known algorithm for the general case runs, as already remarked, in exponential time, and there is no 2 o(n) algorithm under ETH due to classical reductions for HAMILTONIAN CYCLE [5, Theorem 14.6]. EUCLIDEAN TSP, on the other hand, is solvable in subexponential time. For the planar case this has been shown in the early 1990s by Kann [14] and independently by Hwang, Chang and Lee [12], who presented an algorithm with an n O( √ n) running time. Both algorithms use a divide-and-conquer approach that relies on finding a suitable separator. The approach taken by Hwang, Chang and Lee is based on considering a triangulation of the point set such that all segments of the tour appear in the triangulation, and then observing that the resulting planar graph has a separator of size O( √ n). Such a separator can be guessed in n O( √ n) ways, leading to a recursive algorithm with n O( √ n) running time. It seems hard to extend this approach to higher dimensions. Kann obtains his separator in a more geometric way, using the fact that in an optimal tour, there cannot be too many long edges that are relatively close together-see the Packing Property we formulate in Section 2. This makes it possible to compute a separator that is crossed by O( √ n) edges of an optimal tour, which can be guessed in n O( √ n)
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