14 research outputs found
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
Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree
The basic goal of survivable network design is to build a cheap network that
maintains the connectivity between given sets of nodes despite the failure of a
few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of
the most basic problems in this area: given a (-edge)-connected graph
and a set of extra edges (links), select a minimum cardinality subset of
links such that adding to increases its edge connectivity to .
Intuitively, one wants to make an existing network more reliable by augmenting
it with extra edges. The best known approximation factor for this NP-hard
problem is , and this can be achieved with multiple approaches (the first
such result is in [Frederickson and J\'aj\'a'81]).
It is known [Dinitz et al.'76] that CAP can be reduced to the case ,
a.k.a. the Tree Augmentation Problem (TAP), for odd , and to the case ,
a.k.a. the Cactus Augmentation Problem (CacAP), for even . Several better
than approximation algorithms are known for TAP, culminating with a recent
approximation [Grandoni et al.'18]. However, for CacAP the best known
approximation is .
In this paper we breach the approximation barrier for CacAP, hence for
CAP, by presenting a polynomial-time
approximation. Previous approaches exploit properties of TAP that do not seem
to generalize to CacAP. We instead use a reduction to the Steiner tree problem
which was previously used in parameterized algorithms [Basavaraju et al.'14].
This reduction is not approximation preserving, and using the current best
approximation factor for Steiner tree [Byrka et al.'13] as a black-box would
not be good enough to improve on . To achieve the latter goal, we ``open the
box'' and exploit the specific properties of the instances of Steiner tree
arising from CacAP.Comment: Corrected a typo in the abstract (in metadata
Partitioning a call graph
Splitting a large software system into smaller and more manageable units has become an important problem for many organizations. The basic structure of a software system is given by a directed graph with vertices representing the programs of the system and arcs representing calls from one program to another. Generating a good partitioning into smaller modules becomes a minimization problem for the number of programs being called by external programs. First, we formulate an equivalent integer linear programming problem with 0–1 variables. theoretically, with this approach the problem can be solved to optimality, but this becomes very costly with increasing size of the software system. Second, we formulate the problem as a hypergraph partitioning problem. This is a heuristic method using a multilevel strategy, but it turns out to be very fast and to deliver solutions that are close to optimal
An Improved LP-based Approximation for Steiner Tree
The Steiner tree problem is one of the most fundamental-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,Zelikovsky-SIDMA’05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than [Vazirani,Rajagopalan-SODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the�-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components togethe
Submodularity Gaps for Selected Network Design and Matching Problems
Submodularity in combinatorial optimization has been a topic of many studies
and various algorithmic techniques exploiting submodularity of a studied
problem have been proposed. It is therefore natural to ask, in cases where the
cost function of the studied problem is not submodular, whether it is possible
to approximate this cost function with a proxy submodular function.
We answer this question in the negative for two major problems in metric
optimization, namely Steiner Tree and Uncapacitated Facility Location. We do so
by proving super-constant lower bounds on the submodularity gap for these
problems, which are in contrast to the known constant factor cost sharing
schemes known for them. Technically, our lower bounds build on strong lower
bounds for the online variants of these two problems. Nevertheless, online
lower bounds do not always imply submodularity lower bounds. We show that the
problem Maximum Bipartite Matching does not exhibit any submodularity gap,
despite its online variant being only (1 - 1/e)-competitive in the randomized
setting