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.     

On strongly polynomial dual simplex algorithms for the maximum flow problem

Several pivot rules for the dual network simplex algorithm that enable it to solve a maximum flow problem on ann-node,m-arc network in at most 2nm pivots and O(n ² m) time are presented. These rules are based on the concept of apreflow and depend upon the use of node labels which are either the lengths of a shortestpseudoaugmenting path from those nodes to the sink node orvalid underestimates of those lengths. Extended versions of our algorithms are shown to solve an important class of parametric maximum flow problems with no increase in the worst-case pivot and time bounds of these algorithms. This research was supported in part by NSF Grants DMS 91-06195, DMS 94-14438, and CDR 84-21402 and DOE Grant DE-FG02-92ER25126.     

A strongly polynomial algorithm for the transportation problem

For the (linear) transportation problem withm supply nodes,n demand nodes andk feasible arcs we describe an algorithm which runs in time proportional tom logm(k + n logn) (assuming w.l.o.g.mn). The algorithm uses excess scaling. The complexity bound is a slight improvement over the bound achieved by an application of a min-cost-flow algorithm of Orlin to the transportation problem.Corresponding author. Research supported in part by grant no. I-84-095.06/88 of the German—Israeli-Foundation for Scientific Research and Development.     

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.     

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n ² m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.     

An algorithm for the linear complementarity problem with upper and lower bounds

In this paper, we adapt the octahedral simplicial algorithm for solving systems of nonlinear equations to solve the linear complementarity problem with upper and lower bounds. The proposed algorithm generates a piecewise linear path from an arbitrarily chosen pointz ⁰ to a solution point. This path is followed by linear programming pivot steps in a system ofn linear equations, wheren is the size of the problem. The starting pointz ⁰ is left in the direction of one of the 2 ⁿ vertices of the feasible region. The ray along whichz ⁰ is left depends on the sign pattern of the function value atz ⁰. The sign pattern of the linear function and the location of the points in comparison withz ⁰ completely govern the path of the algorithm.This research is part of the VF-Program Equilibrium and Disequilibrium in Demand and Supply, approved by the Netherlands Ministry of Education, Den Haag, The Netherlands.     

Minimum flow problem on network flows with time-varying bounds

In this paper, we consider the minimum flow problem on network flows in which the lower arc capacities vary with time. We will show that this problem for set {0, 1, … , T} of time points can be solved by at most n minimum flow computations, by combining of preflow-pull algorithm and reoptimization techniques (no matter how many values of T are given). Running time of the presented algorithm is O(n²m).     

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.     

Strongly polynomial dual simplex methods for the maximum flow problem

This paper presents dual network simplex algorithms that require at most 2nm pivots and O(n ² m) time for solving a maximum flow problem on a network ofn nodes andm arcs. Refined implementations of these algorithms and a related simplex variant that is not strictly speaking a dual simplex algorithm are shown to have a complexity of O(n ³). The algorithms are based on the concept of apreflow and depend upon the use of node labels that are underestimates of the distances from the nodes to the sink node in the extended residual graph associated with the current flow. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research was supported by NSF Grants DMS 91-06195, DMS 94-14438 and CDR 84-21402 and DOE Grant DE-FG02-92ER25126.Research was supported by NSF Grant CDR 84-21402 at Columbia University.     

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.     

A polynomial dual simplex algorithm for the generalized circulation problem

This paper presents a polynomial-time dual simplex algorithm for the generalized circulation problem. An efficient implementation of this algorithm is given that has a worst-case running time of O(m ²(m+nlogn)logB), where n is the number of nodes, m is the number of arcs and B is the largest integer used to represent the rational gain factors and integral capacities in the network. This running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem. Received: June 1998 / Accepted: June 27, 2001?Published online September 17, 2001     

A faster polynomial algorithm for the unbalanced Hitchcock transportation problem

We present a new algorithm for the Hitchcock transportation problem. On instances with n sources and k sinks, our algorithm has a worst-case running time of O(nk²(logn+klogk)). It closes a gap between algorithms with running time linear in n but exponential in k and a polynomial-time algorithm with running time O(nk²log²n).     

A primal simplex algorithm that solves the maximum flow problem in at mostnm pivots and O(n 2 m) time

We propose a primal network simplex algorithm for solving the maximum flow problem which chooses as the arc to enter the basis one that isclosest to the source node from amongst all possible candidates. We prove that this algorithm requires at mostnm pivots to solve a problem withn nodes andm arcs, and give implementations of it which run in O(n ² m) time. Our algorithm is, as far as we know, the first strongly polynomial primal simplex algorithm for solving the maximum flow problem.This research was supported in part by NSF Grants DMS 85-12277 and CDR 84-21402 and ONR Contract N00014-87-K0214.     

A capacity-rounding algorithm for the minimum-cost circulation problem: A dual framework of the Tardos algorithm

Recently, É. Tardos gave a strongly polynomial algorithm for the minimum-cost circulation problem and solved the open problem posed in 1972 by J. Edmonds and R.M. Karp. Her algorithm runs in O(m ² T(m, n) logm) time, wherem is the number of arcs,n is the number of vertices, andT(m, n) is the time required for solving a maximum flow problem in a network withm arcs andn vertices. In the present paper, taking an approach that is a dual of Tardos's, we also give a strongly polynomial algorithm for the minimum-cost circulation problem. Our algorithm runs in O(m ² S(m, n) logm) time and reduces the computational complexity, whereS(m, n) is the time required for solving a shortest path problem with a fixed origin in a network withm arcs,n vertices, and a nonnegative arc length function. The complexity is the same as that of Orlin's algorithm, recently developed by efficiently implementing the Edmonds-Karp scaling algorithm.     

ON THE COMPUTATIONAL COMPLEXITY OF THE MAXIMUM TRADE PROBLEM

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A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables

We show that the production-transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time. The algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem: monotonicity of the objective function along certain directions, small proportion of nonlinear variables and combinatorial properties implied by transportation constraints.     

A branch and bound algorithm for the maximum clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

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Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds

Ideas of a simplicial variable dimension restart algorithm to approximate zero points onR ⁿ developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds. The algorithm can be considered as a modification of the2n-ray zero point finding algorithm onR ⁿ . It appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n-ray algorithm applied to an equivalent zero point problem. This is caused by the fact that the algorithm utilizes the complementarity conditions on the variables. This work is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply,” which has been approved by the Netherlands Ministry of Education and Sciences.     

A polynomial time algorithm for a chance-constrained single machine scheduling problem

This paper gives an O(n√nlog³n) time algorithm for the chance-constrained sequencing problem on a single machine, where n is the number of jobs and the objective is to minimize the number of jobs which are early with probability not smaller than α (a given constant) against the common due time d.