网络最小覆盖流问题

网络流理论中最基本的模型是最大流及最小费用流问题. 为研 究堵塞现象, 文献中出现了最小饱和流问题, 但它是NP-难的. 研究类似的最小覆盖流问题, 即求一流, 使每一条弧的流量达到一定的额定量, 而流的值为最小. 主要结果是给出多项式时间算法, 并应用于最小饱和流问题.     

A parametric algorithm for convex cost network flow and related problems

The convex cost network flow problem is to determine the minimum cost flow in a network when cost of flow over each arc is given by a piecewise linear convex function. In this paper, we develop a parametric algorithm for the convex cost network flow problem. We define the concept of optimum basis structure for the convex cost network flow problem. The optimum basis structure is then used to parametrize v, the flow to be transsshipped from source to sink. The resulting algorithm successively augments the flow on the shortest paths from source to sink which are implicitly enumerated by the algorithm. The algorithm is shown to be polynomially bounded. Computational results are presented to demonstrate the efficiency of the algorithm in solving large size problems. We also show how this algorithm can be used to (i) obtain the project cost curve of a CPM network with convex time-cost tradeoff functions; (ii) determine maximum flow in a network with concave gain functions; (iii) determine optimum capacity expansion of a network having convex arc capacity expansion costs.     

A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n ²m lognC, n ²m² logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm ² logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).     

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.     

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call theminimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks withn nodes,m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n logn)) time, whereΔ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n ²) pivots and O(nm +n ² logn) time.     

Maximum flow problem on dynamic generative network flows with time-varying bounds

This paper considers a new class of network flows, called dynamic generative network flows in which, the flow commodity is dynamically generated at a source node and dynamically consumed at a sink node and the arc-flow bounds are time dependent. Then the maximum dynamic flow problem in such networks for a pre-specified time horizon T is defined and mathematically formulated in both arc flow and path flow presentations. By exploiting the special structure of the problem, an efficient algorithm is developed to solve the general form of the dynamic problem as a minimum cost static flow problem.     

Imposing Minimax and Quantile Constraints on Optimal Matching in Observational Studies

Modern methods construct a matched sample by minimizing the total cost of a flow in a network, finding a pairing of treated and control individuals that minimizes the sum of within-pair covariate distances subject to constraints that ensure distributions of covariates are balanced. In aggregate, these methods work well; however, they can exhibit a lack of interest in a small number of pairs with large covariate distances. Here, a new method is proposed for imposing a minimax constraint on a minimum total distance matching. Such a match minimizes the total within-pair distance subject to various constraints including the constraint that the maximum pair difference is as small as possible. In an example with 1391 matched pairs, this constraint eliminates dozens of pairs with moderately large differences in age, but otherwise exhibits the same excellent covariate balance found without this additional constraint. A minimax constraint eliminates edges in the network, and can improve the worst-case time bound for the performance of the minimum cost flow algorithm, that is, a better match from a practical perspective may take less time to construct. The technique adapts ideas for a different problem, the bottleneck assignment problem, whose sole objective is to minimize the maximum within-pair difference; however, here, that objective becomes a constraint on the minimum cost flow problem. The method generalizes. Rather than constrain the maximum distance, it can constrain an order statistic. Alternatively, the method can minimize the maximum difference in propensity scores, and subject to doing that, minimize the maximum robust Mahalanobis distance. An example from labor economics is used to illustrate. Supplementary materials for this article are available online.     

Modeling of building evacuation problems by network flows with side constraints

In this paper we model building evacuations by network flows with side constraints. Side constraints come from variable arc capacities on some arcs which are functions of flows in incident arcs. In this context we study maximum flow, minimum cost, and minimax objectives. For some special structured networks we propose ‘greedy’ algorithms for solving these problems. For more general network structures, solution procedures are recommended which take advantage of the network structures of the problems.     

A faster capacity scaling algorithm for minimum cost submodular flow

We describe an O(n ⁴ hmin{logU,n ²logn}) capacity scaling algorithm for the minimum cost submodular flow problem. Our algorithm modifies and extends the Edmonds–Karp capacity scaling algorithm for minimum cost flow to solve the minimum cost submodular flow problem. The modification entails scaling a relaxation parameter δ. Capacities are relaxed by attaching a complete directed graph with uniform arc capacity δ in each scaling phase. We then modify a feasible submodular flow by relaxing the submodular constraints, so that complementary slackness is satisfied. This creates discrepancies between the boundary of the flow and the base polyhedron of a relaxed submodular function. To reduce these discrepancies, we use a variant of the successive shortest path algorithm that augments flow along minimum cost paths of residual capacity at least δ. The shortest augmenting path subroutine we use is a variant of Dijkstra’s algorithm modified to handle exchange capacity arcs efficiently. The result is a weakly polynomial time algorithm whose running time is better than any existing submodular flow algorithm when U is small and C is big. We also show how to use maximum mean cuts to make the algorithm strongly polynomial. The resulting algorithm is the first capacity scaling algorithm to match the current best strongly polynomial bound for submodular flow. Received: August 6, 1999 / Accepted: July 2001?Published online October 2, 2001     

Inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow x⁰, an inverse minimum cost flow problem is to modify the cost vector as little as possible to make x⁰ form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the inverse minimum cost flow problems, where the modification of the arcs is measured by the weighted Hamming distance. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in O(n · m²).     

Finding a feasible flow in a strongly connected network

We give a linear-time algorithm to find a feasible flow in a strongly connected network with fixed supplies and demands, each summing to a common value that is at most the minimum arc capacity. This algorithm speeds up the Goldberg-Rao maximum flow method by a constant factor.     

A specialized network simplex algorithm for the constrained maximum flow problem

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.     

Network simplex algorithm for the general equal flow problem

In this paper the general equal flow problem is considered. This is a minimum cost network flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the same value. This model can be applied to approach water resource system management problems or multiperiod logistic problems in general involving policy restrictions which require some arcs to carry the same amount of flow through the given study period. Although the bases of the general equal flow problem are no longer spanning trees, it is possible to recognize a similar structure that allows us to take advantage of the practical computational capabilities of network models. After characterizing the bases of the problem as good (r+1)-forests, a simplex primal algorithm is developed that exploits the network structure of the problem and requires only slight modifications of the well-known network simplex algorithm.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.     

最小费用最大双流

本文根据一个实例建立了在容量－费用双流网络中求最小费用最大双流的模型，提出了最小费用最大双流和双流增量网络的概念，找出并证明了最小费用双流的充要条件，最后给出该模型的一个算法并估计了算杂性。     

Implementation and Test of Auction Methods for Solving Generalized Network Flow Problems with Separable Convex Cost

We describe the implementation and testing of two methods, based on the auction approach, for solving the problem of minimizing a separable convex cost subject to generalized network flow conservation constraints. The first method is the -relaxation method of Ref. 1; the second is an extension of the auction sequential/shortest path algorithm for ordinary network flow to generalized network flow. We report test results on a large set of randomly generated problems with varying topology, arc gains, and cost function. Comparison with the commercial code CPLEX on linear/quadratic cost problems and with the public-domain code PPRN on nonlinear cost ordinary network problems are also made. The test results show that the auction sequential/shortest path algorithm is generally fastest for solving quadratic cost problems and mixed linear/nonlinear cost problems with arc gain range near 1. The -relaxation method is generally fastest for solving nonlinear cost ordinary network problems and mixed linear/nonlinear cost problems with arc gain range away from 1. CPLEX is generally fastest for solving linear cost and mixed linear/quadratic cost problems with arc gain range near 1.     

Lexicographic optimisation in generalised network flow problems

In this paper the lexicographic optimisation of the multiobjective generalised network flow problem is considered. Optimality conditions are proved on the basis of the equivalence of this problem and a weighted generalised network flow problem. These conditions are used to develop a network-based algorithm which properly modifies primal-dual algorithms for minimum cost generalised network flow problems. Computational results indicate that this algorithm is faster than general-purpose algorithms for linear lexicographic optimisation. Besides, this model is used for approaching a water resource system design problem.     

A logistics problem: a non-markovian queueing network model and algorithm

In this paper, a stochastic version of the classical deterministic balanced single commodity capacitated transportation network problem is presented. In this model, each arc of the network connects a supply node to a demand node and the flow of units forming along each arc of the network forms a stochastic process (i.e.G/M/1 queueing system with generally distributed interarrival time, a Markovian server, a single server, infinite capacity, and the first come first served queueing discipline). In this model, the total transportation cost is minimized such that the total supply rate is equal to the total demand rate, and the resulting probability of finding excessive congestion along each arc (i.e., the resulting probability of finding congestion inside the queueing system formed along each arc in excess of a fixed number) is equal to a desirable value     

On the flow cost lowering problem

This paper presents the flow cost lowering problem (FCLP), which is an extension to the integral version of the well-known minimum cost flow problem (MCFP). While in the MCFP the flow costs are fixed, the FCLP admits lowering the flow cost on each arc by upgrading the arc. Given a flow value and a bound on the total budget which can be used for upgrading the arcs, the goal is to find an upgrade strategy and a flow of minimum cost. The FCLP is shown to be NP-hard even on series–parallel graphs. On the other hand the paper provides a polynomial time approximation algorithm on series–parallel graphs.     

On Solving Quickest Time Problems in Time-Dependent, Dynamic Networks

In this paper, a pseudopolynomial time algorithm is presented for solving the integral time-dependent quickest flow problem (TDQFP) and its multiple source and sink counterparts: the time-dependent evacuation and quickest transshipment problems. A more widely known, though less general version, is the quickest flow problem (QFP). The QFP has historically been defined on a dynamic network, where time is divided into discrete units, flow moves through the network over time, travel times determine how long each unit of flow spends traversing an arc, and capacities restrict the rate of flow on an arc. The goal of the QFP is to determine the paths along which to send a given supply from a single source to a single sink such that the last unit of flow arrives at the sink in the minimum time. The main contribution of this paper is the time-dependent quickest flow (TDQFP) algorithm which solves the TDQFP, i.e. it solves the integral QFP, as defined above, on a time-dependent dynamic network, where the arc travel times, arc and node capacities, and supply at the source vary with time. Furthermore, this algorithm solves the time-dependent minimum time dynamic flow problem, whose objective is to determine the paths that lead to the minimum total time spent completing all shipments from source to sink. An optimal solution to the latter problem is guaranteed to be optimal for the TDQFP. By adding a small number of nodes and arcs to the existing network, we show how the algorithm can be used to solve both the time-dependent evacuation and the time-dependent quickest transshipment problems. This revised version was published online in August 2006 with corrections to the Cover Date.