1,218 research outputs found
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Contraction blockers for graphs with forbidden induced paths.
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pℓ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
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