非二部图中无容量限制和有容量限制的b-匹配问题的近线性时间逼近格式
Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b--Matching Problems in Nonbipartite Graphs
摘要：
我们提出了第一个接近最佳的近似方案的最大加权（无能力或有能力）$b$ -匹配问题的非二分图，运行在时间（近）线性的边缘数。对于任何$δ>3/\sqrt{n}$，算法在$O(m \poly(δ^{-1},\log n))$时间内产生$(1-δ)$近似。我们提供分数的解决方案，这些问题的标准线性规划公式，随后还提供（近）线性时间近似计划舍入的分数的解决方案。通过这些问题作为一个工具，我们还提出了几个想法的上下文中近似使用快速原始-对偶算法求解线性规划。首先，即使这些问题的对偶有指数许多变量和有效的精确计算的对偶权重是不可行的，我们表明，我们可以有效地计算和使用稀疏近似的对偶权重使用的组合（i）添加扰动的约束的多面体和（ii）放大，然后阈值的对偶权重。其次，我们表明，近似算法可以用来减少制定的宽度，更快的收敛。
英文摘要：
We present the first near optimal approximation schemes for the maximum weighted (uncapacitated or capacitated) $b$--matching problems for non-bipartite graphs that run in time (near) linear in the number of edges. For any $δ>3/\sqrt{n}$ the algorithm produces a $(1-δ)$ approximation in $O(m \poly(δ^{-1},\log n))$ 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.
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