Combinatorial algorithms for the generalized circulation problem

Discrete Applied Mathematics, 1981

If f is a function of several variables, one calls a pair of variables substitutes (co.mp/ements) if the change of the value of the function when both variables are increased is at most (at least) equal to the sum of the changes when each is increased separately. We here consider the case wherefis the value of a maximum weight circulation on a network and the variables are the upper and lower bounds and the wei8hts of a pair of arcs. We introduce a simple combinatorial criterion for two arcs to be in "series" or "parallel" and show that these two cases correspond to the variables being complements or substitutes respectively. This generalizes results of Shapley for the special case of the maximum flow and optimal assignment problems. We also show that our result is best possible in that if two arcs are neither in series nor parallel, then the corresponding variables can be either substitutes or complements or both.

Mathematical Programming, 1994

This paper 1 is concerned with generalized network ow problems. In a generalized network, each edge e = ( u v) has a positive \ ow m ultiplier" a e associated with it. The interpretation is that if a ow o f x e enters the edge at node u, then a ow o f a e x e exits the edge at v.

European Journal of Operational Research, 1988

This paper presents a new algorithm for the solution of a network problem with equal flow side constraints. The solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The proposed algorithm makes use of Lagrangean relaxation to obtain a lower bound and decomposition by righ't-hand-side allocation to obtain upper bounds. The Lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound, but also allows an initial allocation of equal flows for the upper bound. The algorithm has been tested on problems with up to 1500 nodes and 6000 arcs. Computational experience indicates that solutions whose objective function value is well within 1% of the optimum can be obtained in 1%-65% of the MPSX time depending on the amount of imbalance inherent in the problem. Incumbent integer solutions which are within 99.99% feasible and well within 1% of the proven lower bound are obtained in a straightforward manner requiring, on the average, 30% of the MPSX time required to obtain a linear optimum.

Networks, 2014

In this article, we propose a generic decomposition scheme for the maximum concurrent flow problem. This decomposition scheme encompasses many models, including, among many others, the classical path formulation and the less studied tree formulation, where the flows of commodities sharing a same source vertex are routed on a set of trees. The pricing problem for this generic model is based on shortest-path computations. We show that the tree-based linear programming formulation can be solved much more quickly than the path or the aggregated arc-flow formulation. Some other decomposition schemes can lead to even faster resolution times. Finally, an efficient strongly polynomial-time combinatorial algorithm is proposed for the single-source case.

Mathematics of Operations Research, 1991

We study the number of pivots required by the primal network simplex algorithm to solve the minimum-cost circulation problem. We propose a pivot selection rule with a.bound-r-° on the number of pivots, for an nvertex network. This is the first known subexponential bound. The network simplex algorithm with this rule can be implemented. to run in n ogn) /2 + 0(l) tine." In the special case of planar graphs, we obtain a polynomial bound on the number of pivots and the running time. We also consider the relaxation of the network simplex algorithm in which cost-increasing pivots are allowed as well as costdecreasing ones. For this algoithm we propose a pivot selection rule with a C" k bound of O(nm ..min {log(KCy, fi6g-l) on the number of pivots, for a network with n vertices, m arcs, and integer arc costs bounded in magnitude by C. The total running time is O(nm logn m min {(lognC), m logn)). This bound is competitive with those of previously known algorithms for the minimum-cost circulation problem.

In many transportation systems, the shipment quantities are subject to minimum lot sizes in addition to regular capacity constraints. This means that either the quantity must be zero, or it must be between the two bounds. In this work, we consider a directed graph, where a minimum lot size and a flow capacity are defined for each arc, and study the problem of maximizing the flow from a given source to a given terminal. We prove that the problem is NP-hard. Based on a straightforward mixed integer programming formulation, we develop a Lagrangean relaxation technique, and demonstrate how this can provide strong bounds on the maximum flow. For fast computation of near-optimal solutions, we develop a heuristic method that departs from the zero solution and gradually augments the set of flow-carrying (open) arcs. The set of open arcs does not necessarily constitute a feasible solution. We point out how feasibility can be checked quickly by solving regular maximum flow problems in an extended network, and how the solutions to these sub-problems can be productive in augmenting the set of open arcs. Finally, we present results from preliminary computational experiments with the construction heuristic.

Journal of Combinatorial Optimization, 2011

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly N P-hard, even if every connected component of the conflict graph is a path of length two. In contrast to this we show that for forcing graphs the problem can be solved efficiently if fractional flow values are allowed. If on the other hand the flow values are required to be integral we provide the sharp line between polynomially solvable and strongly N P-hard instances.

European Journal of Operational Research, 1993

The original motivation for investigating the Linear Balancing Flow Problem (LBFP) came from the optimization of a real-life production plan. Each instance of LBFP is a linear programming problem and can be interpreted as a flow problem on a bipartite network with gains. In the literature the typical flow problem on networks with gains is either a max flow or a rain cost flow problem. Here we consider a different kind of objective, namely, how to optimally balance the flow out of a given set of nodes. An original algorithm is suggested which takes advantage of the particular problem structure. Theoretically it may be viewed as a specialized version of the Simplex. However it is more efficient than the latter: in fact, it requires much less memory and computing time. The key feature consists in associating a tree to the set of variables of the dual problem that are out of the current basis. The justification of the proposed algorithm does not immediately follow from that of the Simplex. A direct proof of its validity is provided and simple numerical examples are reported.

ArXiv, 2021

This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O(F (m+ n) +mn+M) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher’s well-known method for the determination of the K best integer flows [11] at the first sub-optimal flow results in an algorithm with a running time of O(Fm(n logn+m) +M) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so-called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O(Kn3 +M). Besides, we give lower and upper bounds for the number ...

Journal of Combinatorial Theory, Series B, 2000

We pose a new network flow problem and solve it by reducing to the b-matching problem. The result has application to integer multiflow optimization. 2000