52 research outputs found
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
Variants of Plane Diameter Completion
The {\sc Plane Diameter Completion} problem asks, given a plane graph and
a positive integer , if it is a spanning subgraph of a plane graph that
has diameter at most . We examine two variants of this problem where the
input comes with another parameter . In the first variant, called BPDC,
upper bounds the total number of edges to be added and in the second, called
BFPDC, upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in steps.Comment: Accepted in IPEC 201
Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time
AbstractWe give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let bw(G) be the branchwidth of G and gm(G) the largest integer g such that G has a g×g grid as a minor. Let c≥1 be a fixed integer and α,β arbitrary constants satisfying α>c+1 and β>2c+1. We give an algorithm which constructs in O(n1+1clogn) time a branch-decomposition of G with width at most αbw(G). We also give an algorithm which constructs a g×g grid minor of G with g≥gm(G)β in O(n1+1clogn) time. The constants hidden in the Big-O notations are proportional to cα−(c+1) and cβ−(2c+1), respectively
Engineering an Approximation Scheme for Traveling Salesman in Planar Graphs
We present an implementation of a linear-time approximation scheme for the traveling salesman problem on planar graphs with edge weights. We observe that the theoretical algorithm involves constants that are too large for practical use. Our implementation, which is not subject to the theoretical algorithm\u27s guarantee, can quickly find good tours in very large planar graphs
Temporal Separators with Deadlines
We study temporal analogues of the Unrestricted Vertex Separator problem from
the static world. An -temporal separator is a set of vertices whose
removal disconnects vertex from vertex for every time step in a
temporal graph. The -Temporal Separator problem asks to find the minimum
size of an -temporal separator for the given temporal graph. We
introduce a generalization of this problem called the -Temporal
Separator problem, where the goal is to find a smallest subset of vertices
whose removal eliminates all temporal paths from to which take less
than time steps. Let denote the number of time steps over which the
temporal graph is defined (we consider discrete time steps). We characterize
the set of parameters and when the problem is -hard
and when it is polynomial time solvable. Then we present a -approximation
algorithm for the -Temporal Separator problem and convert it to a
-approximation algorithm for the -Temporal Separator problem.
We also present an inapproximability lower bound of for the -Temporal Separator problem assuming that
\mathcal{NP}\not\subset\mbox{\sc Dtime}(n^{\log\log n}). Then we consider
three special families of graphs: (1) graphs of branchwidth at most , (2)
graphs such that the removal of and leaves a tree, and (3) graphs
of bounded pathwidth. We present polynomial-time algorithms to find a minimum
-temporal separator for (1) and (2). As for (3), we show a
polynomial-time reduction from the Discrete Segment Covering problem with
bounded-length segments to the -Temporal Separator problem where the
temporal graph has bounded pathwidth
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