3,140 research outputs found
Augmenting graphs to minimize the diameter
We study the problem of augmenting a weighted graph by inserting edges of
bounded total cost while minimizing the diameter of the augmented graph. Our
main result is an FPT 4-approximation algorithm for the problem.Comment: 15 pages, 3 figure
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
The Traveling Salesman Problem Under Squared Euclidean Distances
Let be a set of points in , and let be a
real number. We define the distance between two points as
, where denotes the standard Euclidean distance between
and . We denote the traveling salesman problem under this distance
function by TSP(). We design a 5-approximation algorithm for TSP(2,2)
and generalize this result to obtain an approximation factor of
for and all .
We also study the variant Rev-TSP of the problem where the traveling salesman
is allowed to revisit points. We present a polynomial-time approximation scheme
for Rev-TSP with , and we show that Rev-TSP is APX-hard if and . The APX-hardness proof carries
over to TSP for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change
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
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
How to Secure Matchings Against Edge Failures
Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems
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