1,679 research outputs found
A LP approximation for the Tree Augmentation Problem
In the Tree Augmentation Problem (TAP) the goal is to augment a tree by a
minimum size edge set from a given edge set such that is
-edge-connected. The best approximation ratio known for TAP is . In the
more general Weighted TAP problem, should be of minimum weight. Weighted
TAP admits several -approximation algorithms w.r.t. to the standard cut
LP-relaxation, but for all of them the performance ratio of is tight even
for TAP. The problem is equivalent to the problem of covering a laminar set
family. Laminar set families play an important role in the design of
approximation algorithms for connectivity network design problems. In fact,
Weighted TAP is the simplest connectivity network design problem for which a
ratio better than is not known. Improving this "natural" ratio is a major
open problem, which may have implications on many other network design
problems. It seems that achieving this goal requires finding an LP-relaxation
with integrality gap better than , which is a long time open problem even
for TAP. In this paper we introduce such an LP-relaxation and give an algorithm
that computes a feasible solution for TAP of size at most times the
optimal LP value. This gives some hope to break the ratio for the weighted
case. Our algorithm computes some initial edge set by solving a partial system
of constraints that form the integral edge-cover polytope, and then applies
local search on -leaf subtrees to exchange some of the edges and to add
additional edges. Thus we do not need to solve the LP, and the algorithm runs
roughly in time required to find a minimum weight edge-cover in a general
graph.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279
LP-Relaxations for Tree Augmentation
In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2-edge-connected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LP-relaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7/4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case
Matching Augmentation via Simultaneous Contractions
We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a 2-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of 13/8 = 1.625 improving upon an earlier 5/3-approximation. The improvement builds on a new ?-approximation preserving reduction for any ? ? 3/2 from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted
Fast Distributed Approximation for TAP and 2-Edge-Connectivity
The tree augmentation problem (TAP) is a fundamental network design problem,
in which the input is a graph and a spanning tree for it, and the goal
is to augment with a minimum set of edges from , such that is 2-edge-connected.
TAP has been widely studied in the sequential setting. The best known
approximation ratio of 2 for the weighted case dates back to the work of
Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was
given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs
give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018],
and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017;
Fiorini et al., SODA 2018].
In this paper, we provide the first fast distributed approximations for TAP.
We present a distributed -approximation for weighted TAP which completes in
rounds, where is the height of . When is large, we show a
much faster 4-approximation algorithm for the unweighted case, completing in
rounds, where is the number of vertices and is
the diameter of .
Immediate consequences of our results are an -round 2-approximation
algorithm for the minimum size 2-edge-connected spanning subgraph, which
significantly improves upon the running time of previous approximation
algorithms, and an -round 3-approximation
algorithm for the weighted case, where is the height of the MST of
the graph. Additional applications are algorithms for verifying
2-edge-connectivity and for augmenting the connectivity of any connected
spanning subgraph to 2.
Finally, we complement our study with proving lower bounds for distributed
approximations of TAP
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