1,976 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
The Traveling Salesman Problem Under Squared Euclidean Distances
Let be a set of points in , and let be a
real number. We define the distance between two points as
, where denotes the standard Euclidean distance between
and . We denote the traveling salesman problem under this distance
function by TSP(). We design a 5-approximation algorithm for TSP(2,2)
and generalize this result to obtain an approximation factor of
for and all .
We also study the variant Rev-TSP of the problem where the traveling salesman
is allowed to revisit points. We present a polynomial-time approximation scheme
for Rev-TSP with , and we show that Rev-TSP is APX-hard if and . The APX-hardness proof carries
over to TSP for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
Approximating Subdense Instances of Covering Problems
We study approximability of subdense instances of various covering problems
on graphs, defined as instances in which the minimum or average degree is
Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We
design new approximation algorithms as well as new polynomial time
approximation schemes (PTASs) for those problems and establish first
approximation hardness results for them. Interestingly, in some cases we were
able to prove optimality of the underlying approximation ratios, under usual
complexity-theoretic assumptions. Our results for the Vertex Cover problem
depend on an improved recursive sampling method which could be of independent
interest
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