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Implementation of approximation algorithms for weighted and unweighted edge-disjoint paths in bidirected trees
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Given a set of weighted directed paths in a bidirected tree, the maximum weight edge-disjoint paths problem (MWEDP) is to select a subset of the given paths such that the selected paths are edge-disjoint and the total weight of the selected paths is maximized. MWEDP is
NP
- hard for bidirected trees of unbounded degree, even if all weights are the same (the unweighted case). Three different approximation algorithms are implemented: a known combinatorial (5/3 + ε)-approximation algorithm
A
1
for the unweighted case, a new combinatorial 2-approximation algorithm
A
2
for the weighted case, and a known (5/3 + ε)-approximation algorithm
A
3
for the weighted case that is based on linear programming. For algorithm
A
1
, it is shown how efficient data structures can be used to obtain a worst-case running-time of
O(m + n + 4
1/ε
√n ċ m)
for instances consisting of
m
paths in a tree with
n
nodes. Experimental results regarding the running-times and the quality of the solutions obtained by the three approximation algorithms are reported. Where possible, the approximate solutions are compared to the optimal solutions, which were computed by running CPLEX on an integer linear programming formulation of MWEDP.
Association for Computing Machinery (ACM)
Title: Implementation of approximation algorithms for weighted and unweighted edge-disjoint paths in bidirected trees
Description:
Given a set of weighted directed paths in a bidirected tree, the maximum weight edge-disjoint paths problem (MWEDP) is to select a subset of the given paths such that the selected paths are edge-disjoint and the total weight of the selected paths is maximized.
MWEDP is
NP
- hard for bidirected trees of unbounded degree, even if all weights are the same (the unweighted case).
Three different approximation algorithms are implemented: a known combinatorial (5/3 + ε)-approximation algorithm
A
1
for the unweighted case, a new combinatorial 2-approximation algorithm
A
2
for the weighted case, and a known (5/3 + ε)-approximation algorithm
A
3
for the weighted case that is based on linear programming.
For algorithm
A
1
, it is shown how efficient data structures can be used to obtain a worst-case running-time of
O(m + n + 4
1/ε
√n ċ m)
for instances consisting of
m
paths in a tree with
n
nodes.
Experimental results regarding the running-times and the quality of the solutions obtained by the three approximation algorithms are reported.
Where possible, the approximate solutions are compared to the optimal solutions, which were computed by running CPLEX on an integer linear programming formulation of MWEDP.
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