14,445 research outputs found
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
Distributed Minimum Cut Approximation
We study the problem of computing approximate minimum edge cuts by
distributed algorithms. We use a standard synchronous message passing model
where in each round, bits can be transmitted over each edge (a.k.a.
the CONGEST model). We present a distributed algorithm that, for any weighted
graph and any , with high probability finds a cut of size
at most in
rounds, where is the size of the minimum cut. This algorithm is based
on a simple approach for analyzing random edge sampling, which we call the
random layering technique. In addition, we also present another distributed
algorithm, which is based on a centralized algorithm due to Matula [SODA '93],
that with high probability computes a cut of size at most
in rounds for any .
The time complexities of both of these algorithms almost match the
lower bound of Das Sarma et al. [STOC '11], thus
leading to an answer to an open question raised by Elkin [SIGACT-News '04] and
Das Sarma et al. [STOC '11].
Furthermore, we also strengthen the lower bound of Das Sarma et al. by
extending it to unweighted graphs. We show that the same lower bound also holds
for unweighted multigraphs (or equivalently for weighted graphs in which
bits can be transmitted in each round over an edge of weight ),
even if the diameter is . For unweighted simple graphs, we show
that even for networks of diameter , finding an -approximate minimum cut
in networks of edge connectivity or computing an
-approximation of the edge connectivity requires rounds
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
Approximating Directed Steiner Problems via Tree Embedding
In the k-edge connected directed Steiner tree (k-DST) problem, we are given a
directed graph G on n vertices with edge-costs, a root vertex r, a set of h
terminals T and an integer k. The goal is to find a min-cost subgraph H of G
that connects r to each terminal t by k edge-disjoint r,t-paths. This problem
includes as special cases the well-known directed Steiner tree (DST) problem
(the case k = 1) and the group Steiner tree (GST) problem. Despite having been
studied and mentioned many times in literature, e.g., by Feldman et al.
[SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit
[SODA'14], there was no known non-trivial approximation algorithm for k-DST for
k >= 2 even in the special case that an input graph is directed acyclic and has
a constant number of layers. If an input graph is not acyclic, the complexity
status of k-DST is not known even for a very strict special case that k= 2 and
|T| = 2.
In this paper, we make a progress toward developing a non-trivial
approximation algorithm for k-DST. We present an O(D k^{D-1} log
n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D
layers, which can be extended to a special case of k-DST on "general graphs"
when an instance has a D-shallow optimal solution, i.e., there exist k
edge-disjoint r,t-paths, each of length at most D, for every terminal t. For
the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h),
thus implying an O(log^3 h)-approximation algorithm for DST that runs in
quasi-polynomial-time (due to the height-reduction of Zelikovsky
[Algorithmica'97]). Consequently, as our algorithm works for general graphs, we
obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance
of the k-edge-connected directed Steiner subgraph problem, where we wish to
connect every pair of terminals by k-edge-disjoint paths
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
Spider covers for prize-collecting network activation problem
In the network activation problem, each edge in a graph is associated with an
activation function, that decides whether the edge is activated from
node-weights assigned to its end-nodes. The feasible solutions of the problem
are the node-weights such that the activated edges form graphs of required
connectivity, and the objective is to find a feasible solution minimizing its
total weight. In this paper, we consider a prize-collecting version of the
network activation problem, and present first non- trivial approximation
algorithms. Our algorithms are based on a new LP relaxation of the problem.
They round optimal solutions for the relaxation by repeatedly computing
node-weights activating subgraphs called spiders, which are known to be useful
for approximating the network activation problem
Approximating ATSP by Relaxing Connectivity
The standard LP relaxation of the asymmetric traveling salesman problem has
been conjectured to have a constant integrality gap in the metric case. We
prove this conjecture when restricted to shortest path metrics of node-weighted
digraphs. Our arguments are constructive and give a constant factor
approximation algorithm for these metrics. We remark that the considered case
is more general than the directed analog of the special case of the symmetric
traveling salesman problem for which there were recent improvements on
Christofides' algorithm.
The main idea of our approach is to first consider an easier problem obtained
by significantly relaxing the general connectivity requirements into local
connectivity conditions. For this relaxed problem, it is quite easy to give an
algorithm with a guarantee of 3 on node-weighted shortest path metrics. More
surprisingly, we then show that any algorithm (irrespective of the metric) for
the relaxed problem can be turned into an algorithm for the asymmetric
traveling salesman problem by only losing a small constant factor in the
performance guarantee. This leaves open the intriguing task of designing a
"good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio
- …