Combinatorial Algorithms for the Unsplittable Flow Problem

We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.     

Simple On-Line Algorithms for the Maximum Disjoint Paths Problem

In this paper we study the classical problem of finding disjoint paths in graphs. This problem has been studied by a number of authors both for specific graphs and general classes of graphs. Whereas for specific graphs many (almost) matching upper and lower bounds are known for the competitiveness of on-line algorithms, not much is known about how well on-line algorithms can perform in the general setting. The best results obtained so far use the expansion of a network to measure the algorithms performance. We use a different parameter called the routing number that, as we will show, allows more precise results than the expansion. It enables us to prove tight upper and lower bounds for deterministic on-line algorithms. The upper bound is obtained by surprisingly simple greedy-like algorithms. Interestingly, our upper bound on the competitive ratio is even better than the best previous approximation ratio for off-line algorithms. Furthermore, we introduce a refined variant of the routing number and show that this variant allows us, for some classes of graphs, to construct on-line algorithms with a competitive ratio significantly below the best possible upper bound that could be obtained using the routing number or the expansion of a network only. We also show that our on-line algorithms can be transformed into efficient algorithms for the more general unsplittable flow problem.     

分层法求解网络最大流的研究

网络最大流问题是经典的组合优化问题,随着网络规模的增加,提高算法效率成为解决问题的关键.为了降低求解大规模网络最大流的计算量,针对单源单汇网络提出基于网络分层的最大流问题求解新方法.分层法首先构造原有向网络对应的层次网络,接着在构造出的层次网络中计算各相邻结点层之间的最大流,以此为基础最终获得整个网络最大流的快速估算.分层法有效降低了计算的复杂性,为在大规模网络中快速获取最大流的求解提供了方便,并给出了一个解决最大流问题的新思路.不同网络上测试的实验结果显示,最大流的近似解误差可控制在1%左右,而平均运行时间仅为经典算法(Ford-Fulkerson算法)运行时间的11%,最好情况下的运行时间仅为经典算法运行时间的2%,是two-phase capacity scaling改进算法运行时间的25%,表明分层方法的有效性.     

Approximation Algorithms for a Capacitated Network Design Problem

We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to support the flow at minimum cost. Capacity can be purchased only in multiples of a fixed quantity. All the flow from a source must be routed in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more effectively models the applications traditionally formulated as capacitated tree problems. We present an approximation algorithm with performance ratio (ρ_ST + 2) where ρ_ST is the performance ratio of any approximation algorithm for the minimum Steiner tree problem. When all sources have unit demand, the ratio improves to ρ_ST + 1) and, in particular, to 2 when all nodes in the graph are sources.     

Approximation Algorithms for a Capacitated Network Design Problem

We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to support the flow at minimum cost. Capacity can be purchased only in multiples of a fixed quantity. All the flow from a source must be routed in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more effectively models the applications traditionally formulated as capacitated tree problems. We present an approximation algorithm with performance ratio (_ST + 2) where _ST is the performance ratio of any approximation algorithm for the minimum Steiner tree problem. When all sources have unit demand, the ratio improves to _ST + 1) and, in particular, to 2 when all nodes in the graph are sources.     

Minimum-cost single-source 2-splittable flow

In the single-source unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain sinks in a given directed graph with edge capacities and costs. The demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover the cost of the solution should not exceed a given budget. An important open question is whether a simultaneous (2,1)-approximation can be achieved for minimizing congestion and cost, i.e., the budget constraint should not be violated. In this note we show that this is possible for the case of 2-splittable flows, i.e., flows where the demand of each commodity is routed along at most two paths. Our result holds under the “no-bottleneck” assumption, i.e., the maximum demand does not exceed the minimum capacity.     

最小权度的网络构建问题

网络构建问题是组合最优化中的经典问题．而连通性是网络设计问题中的一个核心问题。考虑这样一个最优化问题：给定无向图G=（V,E;W）,W：E→Q^＋是权重函数,G’=（V,E’）为G的一个子图．要寻找E的一个子集E”E．使得由E’∪E”所得的诱导子图是一个连通图,其目标是使得所有方案中权度最大者的权度值达到最小。经过对问题分析．对问题的特殊情况E’=Ф,设计了两个时间复杂度分别为O（n^2）和O（mn）的启发式算法。而E”≠Ф的情况也可以类似讨论．     

Steiner Tree问题的研究进展

Steiner树问题是经典的NP难解问题,在计算机网络布局、电路设计以及生物网络等领域都有很多应用。随着参数计算理论的发展,已经证明了无向图和有向图中的Steiner树问题都是固定参数可解的(FPT)。介绍了无向图和有向图中Steiner树问题的近似算法和参数算法,分析了一些特殊Steiner树问题的研究现状,还讨论了顶点加权Steiner树问题的研究进展。最后,提出了该问题的进一步研究方向。     

多处理器任务调度问题的新近似算法

研究任务有多种处理方式的多处理器任务调度问题（MTS）的求解算法，给出求解这种问题的二阶段方法：第1阶段为指派问题，第二阶段调度问题Pm｜fixj｜Cmax，从而得到一个新的求解Pm｜setj｜Cmax。近似算法的方法，并针对P4｜fixj｜Cmax给出了具体算法，证明这种近似算法是一个2-逼近度算法，是文献中在4-处理器问题上的推广。     

一个新的最大流问题增载轨算法

通过放松Ahujia和Orlin算法的约束，给出了一个新的增载轨算法．该算法实质上提供了一个构造、阻塞无环网络的策略，它可以在每次构造无环网络中得到更多的增载轨．从而进一步降低了找到每条增载轨的代价．实验表明，新的算法比Dinic算法快2～5倍，和目前实验性能最好的预流推进算法基本相近．说明增载轨类算法在实际性能方面未必落后于预流推进类算法．