Graphic matroids and the multicommodity transportation problem

A necessary and sufficient condition for unimodularity in the multicommodity transportation problem is established, and the constructive proof yields an equivalent, single commodity network flow problem for the class of problems satisfying the condition. The concept of a graphic matroid is used to establish the transformation.     

Speeding up Karmarkar's algorithm for multicommodity flows

We show how to speed up Karmarkar's linear programming algorithm for the case of multicommodity flows. The special structure of the constraint matrix is exploited to obtain an algorithm for the multicommodity flow problem which requires O(s ^3.5 v ^2.5 eL) arithmetic operations, each operation being performed to a precision of O (L) bits. Herev is the number of vertices ande is the number of edges in the given network,s is the number of commodities, andL is bounded by the number of bits in the input. We obtain a speed up of the order of (e ^0.5/v ^0.5)+(e ^2.5/v ^2.5s²) over Karmarkar's modified algorithm which is substantial for dense networks. The techniques in the paper can also be used to speed up any interior point algorithm for any linear programming problem whose constraint matrix is structurally similar to the one in the multicommodity flow problem. Research supported by a fellowship from the Shell Foundation. Research supported by NSF under grant NSF DCR-8404239.     

Fast deterministic approximation for the multicommodity flow problem

In this paper we consider an optimization version of the multicommodity flow problem which is known as the maximum concurrent flow problem. We show that an approximate solution to this problem can be computed deterministically using O(k(ε ⁻² + logk) logn) 1-commodity minimum-cost flow computations, wherek is the number of commodities,n is the number of nodes, andε is the desired precision. We obtain this bound by proving that in the randomized algorithm developed by Leighton et al. (1995) the random selection of commodities can be replaced by the deterministic round-robin without increasing the total running time. Our bound significantly improves the previously known deterministic upper bounds and matches the best known randomized upper bound for the approximation concurrent flow problem. A preliminary version of this paper appeared inProceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, San Francisco CA, 1995, pp. 486–492.     

Uniform multicommodity flow through the complete graph with random edge-capacities

Give random capacities C to the edges of the complete n-vertex graph. Consider the maximum flow Φ_n that can be simultaneously routed between each source-destination pair. We prove that Φ_n→? in probability where the limit constant ? depends on the distribution of C in a simple way, and that asymptotically one need use only 1- and 2-step routes. The proof uses a reduction to a random graph problem.     

Approximate decomposition methods for the analysis of multicommodity flow routing in generalized queuing networks

In this study we deal with network routing decisions and approximate performance evaluation approaches for generalized open queuing networks (OQN), in which commodities enter the network, receive service at one or more arcs and then leave the network. Exact performance evaluation has been applied for the analysis of Jackson OQN, where the arrival and service processes of the commodities are assumed to be Poisson. However, the Poisson processes’ hypotheses are not a plausible or acceptable assumption for the analysis of generalized OQN, as their arrival and service processes can be much less variable than Poisson processes, resulting in overestimated system performance measures and inappropriate flow routing solutions. In this paper we merge network routing algorithms and network decomposition methods to solve multicommodity flow problems in generalized OQN. Our focus is on steady-state performance measures as average delays and waiting times in queue. The main contributions are twofold: (i) to highlight that solving the corresponding multicommodity flow problem by representing the generalized OQN as a Jackson OQN may be a poor approximation and may lead to inaccurate estimates of the system performance measures, and (ii) to present a multicommodity flow algorithm based on a routing step and on an approximate decomposition step, which leads to much more accurate solutions. Computational results are presented in order to show the effectiveness of the proposed approach.     

Mechanism design for a multicommodity flow game in service network alliances

We study a collaborative multicommodity flow game where individual players own capacity on the edges of the network and share this capacity to deliver commodities. We present membership mechanisms, by adopting a rationality based approach using notions from game theory and inverse optimization, to allocate benefits among the players in such a game.     

On equivalent representations of certain multicommodity networks as single commodity flow problems

Several classes of multicommodity networks have been shown to have the property that they can be transformed to equivalent uncapacitated single commodity flow problems. We show that many of these networks can be further reduced to smaller, semi-capacitated flow problems using the inverse of a result of Ford and Fulkerson. This appears to be a useful computationally-oriented tool for developing practically efficient algorithms. These concepts are also used to establish a generalization of a previous result concerning multicommodity transportation problems.     

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.     

0-1 reformulations of the multicommodity capacitated network design problem

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.     

A cycle augmentation algorithm for minimum cost multicommodity flows on a ring

Minimum cost multicommodity flows are a useful model for bandwidth allocation problems. These problems are arising more frequently as regional service providers wish to carry their traffic over some national core network. We describe a simple and practical combinatorial algorithm to find a minimum cost multicommodity flow in a ring network. Apart from 1 and 2-commodity flow problems, this seems to be the only such “combinatorial augmentation algorithm” for a version of exact mincost multicommodity flow. The solution it produces is always half-integral, and by increasing the capacity of each link by one, we may also find an integral routing of no greater cost. The “pivots” in the algorithm are determined by choosing an >0, increasing and decreasing sets of variables, and adjusting these variables up or down accordingly by . In this sense, it generalizes the cycle cancelling algorithms for (single source) mincost flow. Although the algorithm is easily stated, proof of its correctness and polynomially bounded running time are more complex.     

Computational study of a family of mixed-integer quadratic programming problems

We present computational experience with a branch-and-cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest real-life problems in a few minutes of run-time.     

Using mixed-integer programming to solve power grid blackout problems

We consider optimization problems related to the prevention of large-scale cascading blackouts in power transmission networks subject to multiple scenarios of externally caused damage. We present computation with networks with up to 600 nodes and 827 edges, and many thousands of damage scenarios.     

Computational experience with an interior point algorithm on the satisfiability problem

We apply the zero-one integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.     

The fleet assignment problem: Solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.This work was supported by NSF and AFORS grant DDM-9115768 and NSF grant SES-9122674.Corresponding author.     

Computational experience with bank-one positive definite quasi-newton algorithms

Rank-one positive definite Quasi-Newton algorithms for unconstrained minimization of the type described by Kleinmichel and Spedicato are numerically evaluated versus the BFS algorithm. Results show that, while on certain functions some rank-one methods perform better, overall the BFS still comes first, its superiority being more evident for larger dimensional problems.     

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x ^* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x^*)-LB)/{f(x^*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.