A unified framework for primal-dual methods in minimum cost network flow problems

We introduce a broad class of algorithms for finding a minimum cost flow in a capacitated network. The algorithms are of the primal-dual type. They maintain primal feasibility with respect to capacity constraints, while trying to satisfy the conservation of flow equation at each node by means of a wide variety of procedures based on flow augmentation, price adjustment, and ascent of a dual functional. The manner in which these procedures are combined is flexible thereby allowing the construction of algorithms that can be tailored to the problem at hand for maximum effectiveness. Particular attention is given to methods that incorporate features from classical relaxation procedures. Experimental codes based on these methods outperform by a substantial margin the fastest available primal-dual and primal simplex codes on standard benchmark problems.This work was supported by the National Science Foundation under Contract NSF/ECS 8217668.     

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.     

A strongly polynomial algorithm for the minimum maximum flow degree problem

Let us consider a network flow respecting arc capacities and flow conservation constraints. The flow degree of a node is sum of the flow entering and leaving it. We study the problem of determining a flow that minimizes the maximum flow degree of a node. We show how to solve it in strongly polynomial time with linear programming.     

A note on the minimum interval cost flow problem

We reduce the problem of minimum interval cost flow problem (MICFP) introduced by Hashemi et al. (2006) to the well-known minimum cost flow problem (MCFP).     

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 polynomial algorithm for minimum quadratic cost flow problems

Network flow problems with quadratic separable costs appear in a number of important applications such as; approximating input-output matrices in economy; projecting and forecasting traffic matrices in telecommunication networks; solving nondifferentiable cost flow problems by subgradient algorithms. It is shown that the scaling technique introduced by Edmonds and Karp (1972) in the case of linear cost flows for deriving a polynomial complexity bound for the out-of-kilter method, may be extended to quadratic cost flows and leads to a polynomial algorithm for this class of problems. The method may be applied to the solution of singly constrained quadratic programs and thus provides an alternative approach to the polynomial algorithm suggested by Helgason, Kennington and Lall (1980).     

A new strongly polynomial dual network simplex algorithm

This paper presents a new dual network simplex algorithm for the minimum cost network flow problem. The algorithm works directly on the original capacitated network and runs in O(mn(m +n logn) logn) time for the network withn nodes andm arcs. This complexity is better than the complexity of Orlin, Plotkin and Tardos’ (1993) dual network simplex algorithm by a factor ofm/n.     

A warm-start dual simplex solution algorithm for the minimum flow networks with postoptimality analyses

This paper presents an algorithm for finding the minimum flow in general (s, t) networks with m directed arcs. The minimum flow problem (MFP) arises in many transportation and communication systems. Here, we construct a linear programming (LP) formulation of MFP and develop a simplex-type algorithm to find a minflow. Unlike other simplex-like algorithms, the algorithm developed here starts with an incomplete Basic Variable Set (BVS) initially and then fills-up the BVS completely while pushing toward an optimal vertex. If this results in pushing too far into infeasibility, the next step pulls the solution back to feasibility. Both phases use the Gauss-Jordan pivoting transformation used in the standard simplex and dual simplex algorithms.

The proposed approach has some common features with Dantzig's self-dual simplex algorithm. We have avoided, however, the need for extra variables (slack and surplus) for equality constraints, as well as an artificial constraint for the self-dual algorithm for initial phase and the dual simplex, respectively. The proposed Push phase takes at most 2m − 1 iterations to complete a tree (this augmentation may not be feasible). An infeasible path to the optimal solution contains at most 2m − 1 iterations; therefore in theory, the algorithm needs a sequence of at most 4m − 2 iterations.

Furthermore, the algorithm developed here makes available the full power of LP perturbation analysis and facilitates introducing network structural changes and side constraints also. It can also detect clerical errors in data entry which may make the problem infeasible or unbounded. It is assumed that the reader is familiar with LP terminology.     

A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ^*/logΔ^*) of the optimum in O(mnlogn)-time, where Δ^* is the minimum k such that there is a spanning tree of the graph with maximum degree k.     

The complexity of a minimum reload cost diameter problem

We consider the minimum diameter spanning tree problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: Minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85]. In this model an undirected edge-coloured graph G is given, together with a nonnegative symmetrical integer matrix R specifying the costs of changing from a colour to another one. The reload cost of a path in G arises at its internal nodes, when passing from the colour of one incident edge to the colour of the other. We prove that, unless P=NP, the problem of finding a spanning tree of G having a minimum diameter with respect to reload costs, when restricted to graphs with maximum degree 4, cannot be approximated within any constant α<2 if the reload costs are unrestricted, and cannot be approximated within any constant β<5/3 if the reload costs satisfy the triangle inequality. This solves a problem left open by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85].     

A generic auction algorithm for the minimum cost network flow problem

In this paper we broadly generalize the assignment auction algorithm to solve linear minimum cost network flow problems. We introduce a generic algorithm, which contains as special cases a number of known algorithms, including the -relaxation method, and the auction algorithm for assignment and for transportation problems. The generic algorithm can serve as a broadly useful framework for the development and the complexity analysis of specialized auction algorithms that exploit the structure of particular network problems. Using this framework, we develop and analyze two new algorithms, an algorithm for general minimum cost flow problems, called network auction, and an algorithm for thek node-disjoint shortest path problem.     

A non-cooperative approach to the cost spanning tree problem

We associate to each cost spanning tree problem a non-cooperative game, which is inspired by a real-life problem. We study the Nash equilibria and subgame perfect Nash equilibria of this game. We prove that these equilibria are closely related with situations where agents connect sequentially to the source.Finicial support from the Ministerio de Ciencia y Tecnologia and FEDER, and Xunta de Galicia through grants BEC2002-04102-C02-01 and PGIDIT03PXIC30002PN is gratefully acknowledged.     

A bad network problem for the simplex method and other minimum cost flow algorithms

For any integern, a modified transportation problem with 2n + 2 nodes is constructed which requires 2 ⁿ + 2 ^n–2–2 iterations using all but one of the most commonly used minimum cost flow algorithms.As a result, the Edmonds—Karp Scaling Method [3] becomes the only known good (in the sense of Edmonds) algorithm for computing minimum cost flows.     

Multiple objective minimum cost flow problems: A review

In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem.     

Practical extensions to a minimum cost flow model for level of repair analysis

The level of repair analysis (LORA) gives answers to three questions that are posed when deciding on how to maintain capital goods: (1) which components to repair upon failure and which to discard, (2) at which locations in the repair network to perform each type of repairs, and (3) at which locations in the network to deploy resources, such as test equipment. The goal is to achieve the lowest possible life cycle costs. Various models exist for the LORA problem. However, they tend to be restrictive in that specific business situations cannot be incorporated, such as having repair equipment with finite capacity or the occurrence of unsuccessful repairs or no-fault-founds. We discuss and model such practically relevant extensions to an existing minimum cost flow formulation for the LORA problem. In an extensive numerical experiment, we show that incorporating the model refinements leads to a substantial change in the costs in general. The repair strategy changes substantially only when incorporating finite resource capacities or a probability of unsuccessful repair that is decreasing with an increasing echelon level.     

Simple bounds and greedy algorithms for decomposing a flow into a minimal set of paths

Given arbitrary source and target nodes s, t and an s–t-flow defined by its flow-values on each arc of a network, we consider the problem of finding a decomposition of this flow with a minimal number of s–t-paths. This problem is issued from the engineering of telecommunications networks for which the task of implementing a routing solution consists in integrating a set of end-to-end paths. We show that this problem is NP-hard in the strong sense and give some properties of an optimal solution. We then propose upper and lower bounds for the number of paths in an optimal solution. Finally we develop two heuristics based on the properties of a special set of solutions called saturating solutions.     

An algorithmic study of the Maximum Flow problem: A comparative statistical analysis

In this paper, we show the results of an experimental study about the most important algorithms proposed to solve the Maximum Flow problem. The appropriate statistical analysis not only allows us to justify comparisons between the different procedures but also to obtain classifications of their practical efficiency. Furthermore, an empirical experiment allows us to identify the influence of several parameters that are not included in a theoretical study.     

The folk solution and Boruvka’s algorithm in minimum cost spanning tree problems

Boruvka’s algorithm, which computes a minimum cost spanning tree, is used to define a rule to share the cost among the nodes (agents). We show that this rule coincides with the folk solution, a very well-known rule of this literature.     

A RECOGNITION PROBLEM IN CONVERTING LINEAR PROGRAMMING TO NETWORK FLOW MODELS

The main goal of this paper is to study the following combinatorial problem.given a finite set E={e1,e2,…em} and a subset familly σ={S1,S2,…,Sn}of E,does there exist a tree T with the edge set E such that each induced subgraph T[Si] of Si is precisely a path (1≤i≤k)?