13 research outputs found
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
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems
Moss and Rabani[12] study constrained node-weighted Steiner tree problems
with two independent weight values associated with each node, namely, cost and
prize (or penalty). They give an O(log n)-approximation algorithm for the
prize-collecting node-weighted Steiner tree problem (PCST). They use the
algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm
for the Budgeted node-weighted Steiner tree problem. Their solution may cost up
to twice the budget, but collects a factor Omega(1/log n) of the optimal prize.
We improve these results from at least two aspects.
Our first main result is a primal-dual O(log h)-approximation algorithm for a
more general problem, prize-collecting node-weighted Steiner forest, where we
have (h) demands each requesting the connectivity of a pair of vertices. Our
algorithm can be seen as a greedy algorithm which reduces the number of demands
by choosing a structure with minimum cost-to-reduction ratio. This natural
style of argument (also used by Klein and Ravi[10] and Guha et al.[8]) leads to
a much simpler algorithm than that of Moss and Rabani[12] for PCST.
Our second main contribution is for the Budgeted node-weighted Steiner tree
problem, which is also an improvement to [12] and [8]. In the unrooted case, we
improve upon an O(log^2(n))-approximation of [8], and present an O(log
n)-approximation algorithm without any budget violation. For the rooted case,
where a specified vertex has to appear in the solution tree, we improve the
bicriteria result of [12] to a bicriteria approximation ratio of (1+eps, O(log
n)/(eps^2)) for any positive (possibly subconstant) (eps). That is, for any
permissible budget violation (1+eps), we present an algorithm achieving a
tradeoff in the guarantee for prize. Indeed, we show that this is almost tight
for the natural linear-programming relaxation used by us as well as in [12].Comment: To appear in ICALP 201
Online node-weighted steiner forest and extensions via disk paintings
We give the first polynomial-time online algorithm for the node-weighted Steiner forest problem with a poly-logarithmic competitive ratio. The competitive ratio of our algorithm is optimal up to a logarithmic factor. For the special case of graphs with an excluded fixed minor (e.g., planar graphs), we obtain a logarithmic competitive ratio, which is optimal up to a constant, using a different online algorithm. Both these results are obtained as special cases of generic results for a large class of problems that can be encoded as online {0, 1}-proper functions. Our results are obtained by using a new framework for online network design problems that we call disk paintings. The central idea in this technique is to amortize the cost of primal updates to a set of carefully selected mutually disjoint fixed-radius dual disks centered at a subset of terminals. We hope that this framework will be useful for other online network design problems
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