近似线性时间约简下的两商品流等价于线性规划
Two-Commodity Flow is Equivalent to Linear Programming under Nearly-Linear Time Reductions
摘要：
我们给出了一个近似线性的时间减少，编码任何线性规划作为一个2商品流问题，只有一个小的爆破的大小。在温和的假设下，类似于现代快速求解器的线性规划，我们的减少只会导致一个多对数乘法增加的程序的大小和运行在接近线性的时间。我们的减少适用于高精度的近似算法和精确算法。给定2-商品流问题的近似解，我们可以在线性时间内提取线性规划的解，而误差仅增加多项式因子。这意味着任何解决2-商品流问题的算法都可以在基本相同的时间内解决线性规划。给定一个具有边容量和两个源-汇对的有向图，2-商品流问题的目标是在满足边容量和流守恒的条件下，最大化两个源-汇对之间的流量之和.一个2-商品流可以直接写成线性规划，因此我们在这两类问题之间建立了一个近乎紧等价的关系。我们的证明遵循Itai将线性规划的多项式时间约简为2-商品流问题（JACM'78）的轮廓。Itai的约化证明了2-商品流的精确求解与线性规划的精确求解是多项式时间等价的。我们改进了Itai的减少，几乎保留在每一步的问题表示的大小。此外，我们还建立了近似求解约简中每个中间问题的误差界，并证明了累积误差是多项式有界的。我们注意到，我们的约简不运行在强多项式时间，它是开放的2-商品流和线性规划是否是等价的强多项式时间。
英文摘要：
We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size. Under mild assumptions similar to those employed by modern fast solvers for linear programs, our reduction causes only a polylogarithmic multiplicative increase in the size of the program and runs in nearly-linear time. Our reduction applies to high-accuracy approximation algorithms and exact algorithms. Given an approximate solution to the 2-commodity flow problem, we can extract a solution to the linear program in linear time with only a polynomial factor increase in the error. This implies that any algorithm that solves the 2-commodity flow problem can solve linear programs in essentially the same time. Given a directed graph with edge capacities and two source-sink pairs, the goal of the 2-commodity flow problem is to maximize the sum of the flows routed between the two source-sink pairs subject to edge capacities and flow conservation. A 2-commodity flow can be directly written as a linear program, and thus we establish a nearly-tight equivalence between these two classes of problems. Our proof follows the outline of Itai's polynomial-time reduction of a linear program to a 2-commodity flow problem (JACM'78). Itai's reduction shows that exactly solving 2-commodity flow and exactly solving linear programming are polynomial-time equivalent. We improve Itai's reduction to nearly preserve the problem representation size in each step. In addition, we establish an error bound for approximately solving each intermediate problem in the reduction, and show that the accumulated error is polynomially bounded. We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time.
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