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.     

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).     

Integer equal flows

The integer equal flow problem is an NP-hard network flow problem, in which all arcs in given sets R₁,…,R_ℓ must carry equal flow. We show that this problem is effectively inapproximable, even if the cardinality of each set R_k is two. When ℓ is fixed, it is solvable in polynomial time.     

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 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.     

An overview on the simplex algorithm

In this paper, the simplex algorithm and its variants are investigated. First, we define a new concept called formal tableau, which leads to derive easily the dual solution from the latest primal table; without any distinction between the original variables and the slack ones. Second, we propose a new method for initializing the simplex algorithm. Unlike the two-phase and the big-M methods, our technique does not involve artificial variables. The computational results reveal that this new method is very favorable especially when the number of artificial variables is significant. Finally, this method will be combined with the notion of formal tableau leading naturally to a second new approach.     

A largest-distance pivot rule for the simplex algorithm

We propose a new pivot rule for the simplex algorithm, which is demonstrative in the dual space intuitively. Although it is based on normalized reduced costs, like the steepest-edge rule and its variants, the rule is much simpler and cheaper than the latter. We report computational results obtained with the 47 largest Netlib problems in terms of the number of rows and columns, all of the 16 Kennington problems, and the 17 largest BPMPD problems. Over the total 80 problems, a variant of the rule outperformed the Devex rule with iterations and time ratio 1.43 and 3.24, respectively.     

Integer version of the multipath flow network synthesis problem

We consider the following integer multipath flow network synthesis problem. We are given two positive integers q, n, (1<q<n), and a non-negative, integer, symmetric, n×n matrix R, each non-diagonal element r_ij of which represents the minimum requirement of q-path flow value between nodes i and j in an undirected network on the node set N={1,2,…,n}. We want to construct a simple, undirected network G=[N,E] with integer edge capacities {u_e:e∈E} such that each of these flow requirements can be realized (one at a time) and the sum of all the edge capacities is minimum. We present an O(n³) combinatorial algorithm for the problem and we show that the problem has integer rounding property.     

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     

Diagnosing infeasibilities in network flow problems

We consider the problem of finding a feasible flow in a directed networkG = (N,A) in which each nodei N has a supplyb(i), and each arc(i,j) A has a zero lower bound on flow and an upper boundu _ij . It is well known that this feasibility problem can be transformed into a maximum flow problem. It is also well known that there is no feasible flow in the networkG if and only if there is a subsetS of nodes such that the net supplies of the nodes inS exceeds the capacity of the arcs emanating fromS. Such a setS is called awitness of infeasibility (or, simply, awitness) of the network flow problem. In the case that there are many different witnesses for an infeasible problem, a small cardinality witness may be preferable in practice because it is generally easier for the user to assimilate, and may provide more guidance to the user on how to identify the cause of the infeasibility. Here we show that the problem of finding a minimum cardinality witness is NP-hard. We also consider the problem of determining aminimal witness, that is, a witnessS such that no proper subset ofS is also a witness. In this paper, we show that we can determine a minimal witness by solving a sequence of at mostn maximum flow problems. Moreover, if we use the preflow-push algorithm to solve the resulting maximum flow problems and organize computations properly, then the total time taken by the algorithm is comparable to that of solving a single maximum flow problem. This approach determines a minimal cardinality witness in O(n ² m ^1/2) time using simple data structures and in O(nm logn) time using the standard implementation of the dynamic tree data structures. We also show that the recognition version of the minimal witness problem is equivalent to a recognition version of a related problem known as theminimum rooted cut problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

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.     

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.     

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.     

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.     

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.     

An efficient approach to updating simplex multipliers in the simplex algorithm

The simplex algorithm computes the simplex multipliers by solving a system (or two triangular systems) at each iteration. This note offers an efficient approach to updating the simplex multipliers in conjunction with the Bartels–Golub and Forrest–Tomlin updates for LU factors of the basis. It only solves one triangular system. The approach was implemented within and tested against MINOS 5.51 on 129 problems from Netlib, Kennington and BPMPD. Computational results show that the new approach improves simplex implementations. Project 10371017 supported by National Natural Science Foundation of China.     

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 exterior point simplex algorithm for (general) linear programming problems

We present an exterior point simplex type algorithm that possesses a new monotonic property. A dual feasible basic solution is required to start with. Intermediate solutions are neither primal nor dual feasible. Cycling-free pivoting rules and an exponentional example are presented.