1 research outputs found
Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b--Matching Problems in Nonbipartite Graphs
We present the first near optimal approximation schemes for the
maximum weighted (uncapacitated or capacitated) --matching
problems for non-bipartite graphs that run in time (near) linear in
the number of edges. For any the algorithm
produces a approximation in O(m \poly(\delta^{-1},\log
n)) time. We provide fractional solutions for the standard linear
programming formulations for these problems and subsequently also
provide (near) linear time approximation schemes
for rounding the fractional solutions.
Through these problems as a vehicle, we also present several ideas
in the context of solving linear programs approximately using fast
primal-dual algorithms. First, even though the dual of these
problems have exponentially many variables and an efficient exact
computation of dual weights is infeasible, we show that we can
efficiently compute and use a sparse approximation of the dual
weights using a combination of (i) adding perturbation to the
constraints of the polytope and (ii) amplification followed by
thresholding of the dual weights. Second, we show that
approximation algorithms can be used to reduce the width of the
formulation, and faster convergence