96,479 research outputs found

    Fast Dynamic Pointer Following via Link-Cut Trees

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    In this paper, we study the problem of fast dynamic pointer following: given a directed graph GG where each vertex has outdegree 11, efficiently support the operations of i) changing the outgoing edge of any vertex, and ii) find the vertex kk vertices `after' a given vertex. We exhibit a solution to this problem based on link-cut trees that requires O(lgn)O(\lg n) time per operation, and prove that this is optimal in the cell-probe complexity model.Comment: 7 page

    Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

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    Given a directed graph GG, a set of kk terminals and an integer pp, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set SS of at most pp (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where SS is a set of at most pp edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of kk given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by pp. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by pp. We complete the picture here by our main result which is that both \textsc{Directed Vertex Multiway Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time 22O(p)nO(1)2^{2^{O(p)}}n^{O(1)}, i.e., FPT parameterized by size pp of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that \textsc{Directed Multicut} is FPT for the case of k=2k=2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011)

    A randomized polynomial kernel for Subset Feedback Vertex Set

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    The Subset Feedback Vertex Set problem generalizes the classical Feedback Vertex Set problem and asks, for a given undirected graph G=(V,E)G=(V,E), a set SVS \subseteq V, and an integer kk, whether there exists a set XX of at most kk vertices such that no cycle in GXG-X contains a vertex of SS. It was independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi and Kobayashi (JCTB '12) that Subset Feedback Vertex Set is fixed-parameter tractable for parameter kk. Cygan et al. asked whether the problem also admits a polynomial kernelization. We answer the question of Cygan et al. positively by giving a randomized polynomial kernelization for the equivalent version where SS is a set of edges. In a first step we show that Edge Subset Feedback Vertex Set has a randomized polynomial kernel parameterized by S+k|S|+k with O(S2k)O(|S|^2k) vertices. For this we use the matroid-based tools of Kratsch and Wahlstr\"om (FOCS '12) that for example were used to obtain a polynomial kernel for ss-Multiway Cut. Next we present a preprocessing that reduces the given instance (G,S,k)(G,S,k) to an equivalent instance (G,S,k)(G',S',k') where the size of SS' is bounded by O(k4)O(k^4). These two results lead to a polynomial kernel for Subset Feedback Vertex Set with O(k9)O(k^9) vertices

    Brief Announcement: Bounded-Degree Cut is Fixed-Parameter Tractable

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    In the bounded-degree cut problem, we are given a multigraph G=(V,E), two disjoint vertex subsets A,B subseteq V, two functions u_A, u_B:V -> {0,1,...,|E|} on V, and an integer k >= 0. The task is to determine whether there is a minimal (A,B)-cut (V_A,V_B) of size at most k such that the degree of each vertex v in V_A in the induced subgraph G[V_A] is at most u_A(v) and the degree of each vertex v in V_B in the induced subgraph G[V_B] is at most u_B(v). In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a 2^{18k}|G|^{O(1)}-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems

    Graphs with many independent vertex cuts

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    The cycles are the only 22-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer k3k\ge 3 there exists a unique graph GG satisfying the following conditions: (1) GG is kk-connected; (2) the independence number of GG is greater than k;k; (3) any independent set of cardinality kk is a vertex cut of G.G. The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery
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