7,757 research outputs found
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
Cubic Augmentation of Planar Graphs
In this paper we study the problem of augmenting a planar graph such that it
becomes 3-regular and remains planar. We show that it is NP-hard to decide
whether such an augmentation exists. On the other hand, we give an efficient
algorithm for the variant of the problem where the input graph has a fixed
planar (topological) embedding that has to be preserved by the augmentation. We
further generalize this algorithm to test efficiently whether a 3-regular
planar augmentation exists that additionally makes the input graph connected or
biconnected. If the input graph should become even triconnected, we show that
the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
Edge covering with budget constrains
We study two related problems: finding a set of k vertices and minimum number
of edges (kmin) and finding a graph with at least m' edges and minimum number
of vertices (mvms).
Goldschmidt and Hochbaum \cite{GH97} show that the mvms problem is NP-hard
and they give a 3-approximation algorithm for the problem. We improve
\cite{GH97} by giving a ratio of 2. A 2(1+\epsilon)-approximation for the
problem follows from the work of Carnes and Shmoys \cite{CS08}. We improve the
approximation ratio to 2. algorithm for the problem. We show that the natural
LP for \kmin has an integrality gap of 2-o(1). We improve the NP-completeness
of \cite{GH97} by proving the pronlem are APX-hard unless a well-known instance
of the dense k-subgraph admits a constant ratio. The best approximation
guarantee known for this instance of dense k-subgraph is O(n^{2/9})
\cite{BCCFV}. We show that for any constant \rho>1, an approximation guarantee
of \rho for the \kmin problem implies a \rho(1+o(1)) approximation for \mwms.
Finally, we define we give an exact algorithm for the density version of kmin.Comment: 17 page
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