968 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 of robust spanning tree problems
In this paper the minimum spanning tree problem with uncertain edge costs is
discussed. In order to model the uncertainty a discrete scenario set is
specified and a robust framework is adopted to choose a solution. The min-max,
min-max regret and 2-stage min-max versions of the problem are discussed. The
complexity and approximability of all these problems are explored. It is proved
that the min-max and min-max regret versions with nonnegative edge costs are
hard to approximate within for any unless
the problems in NP have quasi-polynomial time algorithms. Similarly, the
2-stage min-max problem cannot be approximated within unless the
problems in NP have quasi-polynomial time algorithms. In this paper randomized
LP-based approximation algorithms with performance ratio of for
min-max and 2-stage min-max problems are also proposed
Vertex and edge covers with clustering properties: complexity and algorithms
We consider the concepts of a t-total vertex cover and a t-total edge cover (t≥1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NP-completeness and approximability results (both upper and lower bounds) and FTP algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FTP algorithm for the latter problem
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
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