148 research outputs found
Odd Multiway Cut in Directed Acyclic Graphs
We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is an extension of the shadow-removal framework for parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for 2 terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for 2 terminals
Odd multiway cut in directed acyclic graphs
We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is a broadening of the shadow-removal framework to address parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for 2 terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for 2 terminals
Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms
We present two new combinatorial tools for the design of parameterized
algorithms. The first is a simple linear time randomized algorithm that given
as input a -degenerate graph and an integer , outputs an independent
set , such that for every independent set in of size at most ,
the probability that is a subset of is at least .The second is a new (deterministic) polynomial
time graph sparsification procedure that given a graph , a set of terminal pairs and an
integer , returns an induced subgraph of that maintains all
the inclusion minimal multicuts of of size at most , and does not
contain any -vertex connected set of size . In
particular, excludes a clique of size as a
topological minor. Put together, our new tools yield new randomized fixed
parameter tractable (FPT) algorithms for Stable - Separator, Stable Odd
Cycle Transversal and Stable Multicut on general graphs, and for Stable
Directed Feedback Vertex Set on -degenerate graphs, resolving two problems
left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our
algorithms can be derandomized at the cost of a small overhead in the running
time.Comment: 35 page
On the Parameterized Complexity of Simultaneous Deletion Problems
For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, ldots, F_alpha)-Deletion problem. In the latter problem, we are given an alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 leq i leq alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = ldots = F_alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and alpha, and can be solved in O(2^{O(alpha k)}n^{O(1)})
time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, ldots, F_alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i\u27s is the class of bipartite graphs and the rest (if any) are forests.
We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = ldots = F_alpha is the family of all forests then the problem is fixed-parameter tractable
parameterized by k and alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Given an undirected graph , a collection of
pairs of vertices, and an integer , the Edge Multicut problem ask if there
is a set of at most edges such that the removal of disconnects
every from the corresponding . Vertex Multicut is the analogous
problem where is a set of at most vertices. Our main result is that
both problems can be solved in time , i.e.,
fixed-parameter tractable parameterized by the size of the cutset in the
solution. By contrast, it is unlikely that an algorithm with running time of
the form exists for the directed version of the problem, as
we show it to be W[1]-hard parameterized by the size of the cutset
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
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