1,175 research outputs found
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
Improved Inapproximability Results for Maximum k-Colorable Subgraph
We study the maximization version of the fundamental graph coloring problem.
Here the goal is to color the vertices of a k-colorable graph with k colors so
that a maximum fraction of edges are properly colored (i.e. their endpoints
receive different colors). A random k-coloring properly colors an expected
fraction 1-1/k of edges. We prove that given a graph promised to be
k-colorable, it is NP-hard to find a k-coloring that properly colors more than
a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2)
was known. Our result pins down the correct asymptotic dependence of the
approximation factor on k. Along the way, we prove that approximating the
Maximum 3-colorable subgraph problem within a factor greater than 32/33 is
NP-hard. Using semidefinite programming, it is known that one can do better
than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges
in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to
properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of
edges of a k-colorable graph.Comment: 16 pages, 2 figure
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