199 research outputs found
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs
We study the prize-collecting version of the Node-weighted Steiner Tree
problem (NWPCST) restricted to planar graphs. We give a new primal-dual
Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar
NWPCST. We then show a ()-approximation which establishes a
new best approximation guarantee for planar NWPCST. This is done by combining
our LMP algorithm with a threshold rounding technique and utilizing the
2.4-approximation of Berman and Yaroslavtsev for the version without penalties.
We also give a primal-dual 4-approximation algorithm for the more general
forest version using techniques introduced by Hajiaghay and Jain
An approximation algorithm to the k-Steiner Forest problem
AbstractGiven a graph G, an integer k, and a demand set D={(s1,t1),…,(sl,tl)}, the k-Steiner Forest problem finds a forest in graph G to connect at least k demands in D such that the cost of the forest is minimized. This problem was proposed by Hajiaghayi and Jain in SODA’06. Thereafter, using a Lagrangian relaxation technique, Segev et al. gave the first approximation algorithm to this problem in ESA’06, with performance ratio O(n2/3logl). We give a simpler and faster approximation algorithm to this problem with performance ratio O(n2/3logk) via greedy approach, improving the previously best known ratio in the literature
2-Approximation for Prize-Collecting Steiner Forest
Approximation algorithms for the prize-collecting Steiner forest problem
(PCSF) have been a subject of research for over three decades, starting with
the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on
Steiner forest and prize-collecting problems. In this paper, we propose and
analyze a natural deterministic algorithm for PCSF that achieves a
-approximate solution in polynomial time. This represents a significant
improvement compared to the previously best known algorithm with a
-approximation factor developed by Hajiaghayi and Jain in 2006.
Furthermore, K{\"{o}}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have
established an integrality gap of at least for the natural LP relaxation
for PCSF. However, we surpass this gap through the utilization of a
combinatorial algorithm and a novel analysis technique. Since is the best
known approximation guarantee for Steiner forest problem, which is a special
case of PCSF, our result matches this factor and closes the gap between the
Steiner forest problem and its generalized version, PCSF
Node-weighted Steiner tree and group Steiner tree in planar graphs
We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games.
The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group
Approximating Node-Weighted k-MST on Planar Graphs
We study the problem of finding a minimum weight connected subgraph spanning
at least vertices on planar, node-weighted graphs. We give a
(4+\eps)-approximation algorithm for this problem. We achieve this by
utilizing the recent LMP primal-dual -approximation for the node-weighted
prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an
approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation
for the edge-weighted variant. In particular, we improve the procedure of
picking additional vertices (tree merging procedure) given by Sadeghian (2013)
by taking a constant number of recursive steps and utilizing the limited
guessing procedure of Arora and Karakostas (Math.\ Prog.\ '06). More generally,
our approach readily gives a (\nicefrac{4}{3}\cdot r+\eps)-approximation on
any graph class where the algorithm of Byrka et al.\ for the prize-collecting
version gives an -approximation. We argue that this can be interpreted as a
generalization of an analogous result by K\"onemann et al. (Algorithmica~'11)
for partial cover problems. Together with a lower bound construction by Mestre
(STACS'08) for partial cover this implies that our bound is essentially best
possible among algorithms that utilize an LMP algorithm for the Lagrangian
relaxation as a black box. In addition to that, we argue by a more involved
lower bound construction that even using the LMP algorithm by Byrka et al.\ in
a \emph{non-black-box} fashion could not beat the factor \nicefrac{4}{3}\cdot
r when the tree merging step relies only on the solutions output by the LMP
algorithm
Node-Weighted Prize Collecting Steiner Tree and Applications
The Steiner Tree problem has appeared in the Karp's list of the first 21 NP-hard problems and is well known as one of the most fundamental problems in Network Design area. We study the Node-Weighted version of the Prize
Collecting Steiner Tree problem.
In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our
goal is to find a subtree T of the graph minimizing the cost of the
nodes in T plus penalty of the nodes not in T. By a reduction
from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1-o(1))ln n unless NP has quasi-polynomial time algorithms, where n is the number of vertices of the graph.
Moss and Rabani claimed an O(log n)-approximation algorithm for the problem using a Primal-Dual approach in their STOC'01 paper \cite{moss2001}. We show that their algorithm is incorrect by providing a counter example in which there is an O(n) gap between the dual solution constructed by their algorithm and the optimal solution. Further, evidence is given that their algorithm probably does not have a simple fix. We propose a new algorithm which is more involved and
introduces novel ideas in primal dual approach for network design problems. Also, our algorithm is a Lagrangian Multiplier Preserving algorithm and we show how this property can be utilized to design an O(log n)-approximation algorithm for the Node-Weighted Quota Steiner Tree problem
using the Lagrangian Relaxation method.
We also show an application of the Node Weighted Quota Steiner Tree problem in designing algorithm with better approximation factor for
Technology Diffusion problem, a problem proposed by Goldberg and Liu
in \cite{goldberg2012} (SODA 2013). In Technology Diffusion, we are given a graph G and a threshold θ(v) associated with each vertex v and we are seeking a set of initial nodes called the seed set.
Technology Diffusion is a dynamic process defined over time in which each vertex is either active or inactive. The vertices in the seed set
are initially activated and each other vertex v gets activated whenever there are at least θ(v) active nodes connected to
v through other active nodes. The Technology Diffusion problem asks to
find the minimum seed set activating all nodes. Goldberg
and Liu gave an O(rllog n)-approximation algorithm for the problem where
r and l are the diameter of G and the number of distinct threshold values, respectively. We improve the approximation factor
to O(min{r,l}log n) by establishing a close connection between the problem and the Node Weighted Quota Steiner Tree problem
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