A tight max-flow min-cut duality theorem for nonlinear multicommodity flows

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Journal of Combinatorial Optimization Pub Date : 2024-04-11 DOI:10.1007/s10878-024-01120-2
Matthew Broussard, Bala Krishnamoorthy
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Abstract

The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity. We study a multiple commodity generalization of Max-Flow in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in \(\mathbb {R}^k\). Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the mutual capacity of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with m constraints in at most \(\mathbb {R}^{(m-n+1)k}\) space from one that requires mn constraints in \(\mathbb {R}^{mk}\) space for a network with n nodes and m edges. We present efficient algorithms that compute \(\epsilon \)-approximations to both the ratio and the integer ratio maximum flow problems.

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非线性多商品流的紧密最大流最小切对偶定理
Max-Flow Min-Cut 定理是 Max-Flow 问题的经典对偶结果,它考虑的是单一商品的流动。我们研究了 Max-Flow 的多商品一般化,其中流量由实值 k 向量组成,流经由 \(\mathbb {R}^k\) 中的区域形成的弧容量网络。鉴于在多商品情况下没有明确的排序概念,我们将广义最大流定义为所有流值的可行区域。我们定义了多商品环境中流量和切分的一系列概念和操作。我们研究了一组切口的互容性,其定义为可以通过该切口的所有流量的集合。我们提出了一种计算成对切口互容的方法,然后将其推广到任意切口集的计算方法。我们证明,互容正好是网络中可行流量的集合,因此等于最大流量。此外,我们还提出了一类简单的多商品最大流量问题,在这类问题中,使用这种紧密对偶性结果的计算速度明显快于默认的蛮力计算。我们还研究了多商品最大流量问题更易处理的特例,其目标是通过网络传输给定向量的最大实数或整数倍。我们设计了一种增量循环搜索算法,它可以将一个具有 n 个节点和 m 条边的网络优化问题,从一个需要在 \(\mathbb {R}^{(m-n+1)k}\) 空间中设置 mn 个约束的问题,简化为一个在最多 \(\mathbb {R}^{(m-n+1)k}\) 空间中设置 m 个约束的问题。我们提出了计算比率问题和整数比率最大流量问题的 \(\epsilon \)-近似值的高效算法。
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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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