On the simplex algorithm for networks and generalized networks (original) (raw)

A new pivot selection rule for the network simplex algorithm

Mathematical Programming, 1997

We present a new network simplex pivot selection rule, which we call the minimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks with n 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 role 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(nA) pivots and in O(4(m + n logn)) time, where A is any upper bound on the sum of all arc flows in every feasible flow. For assi~ment and shortest path problems, our algorithm runs in O(n 2) pivots and O(nm + n 2 logn) time.

Improved primal simplex algorithms for shortest path, assignment and minimum cost flow problems

In this paper, we present a new primal simplex pivot rule and analyse the worst-case complexity of the resulting simplex algorithm for the minium cost flow problem, the assignment problem and the shortest path problem. We consider networks with n nodes, m arcs, integral arc capacities bounded by an integer number U, and integral arc costs bounded by an integer number C. Let L and U denote the nonbasic arcs at their lower and upper bounds respectively, and cij denote the reduced cost of any arc (i, j). Further, let A be a parameter whose initial value is C. Then our pivot rule is as follows: Select as an entering arc any (i, j) L with cij < -A/2 or any (i, j) U with cij A/2; select the leaving arc so that the strong feasibility of the basis is maintained. When there is no nonbasic arc satisfying this rule then replace A by A/2. We show that the simplex algorithm using this rule performs O(nm U logC) pivots and can be implemented to run in O(m 2 U logC) time. Specializing these results for the assignment and shortest path problems we show that the simplex algorithm solves these problems in O(n 2 logC) pivots and O(nm logC) time. These algorithms use the same data structures that are typically used to implement the primal simplex algorithms for network problems and have enough flexibility for fine tuning the algorithms in practice. We also use these ideas to obtain an O(nm logC) label correcting algorithm for the shortest path problem with arbitrary arc lengths, and an improved implementation of Dantzig's pivot rule.

Genuinely Polynomial Simplex and Non-Simplex Algorithms for the Minimum Cost Network Flow Problem

We consider the minimum cost network flow problem min(cx : Ax=b, x > 0) on a graph G = (V,E). First we give a minor modification of Edmonds-Karp scaling technique, and we show that it solves the minimum cost flow problem in 0((IV1 2 log IVI)(IEI + IVi log VI)) steps. We also provide two dual simplex algorithms that solve the minimum cost flow problem in O(IV1 4 log IVi) pivots and O(IVi 3 log IVi) pivots respectively. Moreover, this latter dual simplex algorithm may be implemented so that the running time is proportional to that of Edmonds-Karp scaling technique.

A network simplex algorithm with O(n) consecutive degenerate pivots

Operations Research Letters, 2002

In this paper, we suggest a new pivot rule for the primal simplex algorithm for the minimum cost ow problem, known as the network simplex algorithm. Due to degeneracy, cycling may occur in the network simplex algorithm. The cycling can be prevented by maintaining strongly feasible bases proposed by Cunningham (Math. Programming 11 (1976) 105; Math. Oper. Res. 4 (1979) 196); however, if we do not impose any restrictions on the entering variables, the algorithm can still perform an exponentially long sequence of degenerate pivots. This phenomenon is known as stalling. Researchers have suggested several pivot rules with the following bounds on the number of consecutive degenerate pivots: m; n 2 ; k(k + 1)=2, where n is the number of nodes in the network, m is the number of arcs in the network, and k is the number of degenerate arcs in the basis. (Observe that k 6 n.) In this paper, we describe an anti-stalling pivot rule that ensures that the network simplex algorithm performs at most k consecutive degenerate pivots. This rule uses a negative cost augmenting cycle to identify a sequence of entering variables.

Genuinely polynomial simplex and non-simplex algorithms for the minimum cost flow problem

Statistics & Probability Letters - STAT PROBAB LETT, 1985

We consider the minimum cost network flow problem min(cx : Ax=b, x > 0) on a graph G = (V,E). First we give a minor modification of Edmonds-Karp scaling technique, and we show that it solves the minimum cost flow problem in,0((IV1, log IVi) pivots respectively.,Moreover, this latter dual simplex algorithm may be implemented so that the running time is proportional to that of Edmonds-Karp,scaling,technique. Key words: network flow, scaling, simplex algorithm, polynomial algorithm. 2

Polynomial-time primal simplex algorithms for the minimum cost network flow problem

Algorithmica, 1992

We present two variants of the primal network simplex algorithm which solve the minimum cost network flow problem in at most O(n2m 2 log n) pivots. Here we define the network simplex method as a method which proceeds from basis tree to adjacent basis tree regardless of the change in objective function value; i.e., the objective function is allowed to increase on some iterations. The first method is an extension of the minimum mean augmenting cycle-canceling method of Goldberg and Tarjan. The second method is a combination of a cost-scaling technique and a primal network simplex method for the maximum flow problem. We also show that the diameter of the primal network flow polytope is at most n2m.

Efficiency of the Network Simplex Algorithm for the Maximum Flow Problem

1988

Goldfarb and Hao have proposed a. network simplex a,lgorithm that will solve a ma.ximum flow problem on an n-vertex, m-arc network in at most nm pivots and 0(n2m) time. In this paper we describe how to implement their algorithm to run in O(nm log n) time by using an extension of the dynamic tree data structure of Sleator and Tarjan. This bound is less than a logarithmic factor la,rger than that of any other known algorithm for the problem.

Polynomial dual network simplex algorithms

Mathematical Programming, 1993

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m 2 log n) bound on the number of pivots, where n and m denotes the number of nodes and arcs in the input network. If the demands are integral and at most B, we also give an O(m(m+n log n) min(lognB; m log n))-time implementation of a strategy that requires somewhat more pivots.

An exterior Simplex type algorithm for the minimum cost network flow problem

Computers & Operations Research, 2009

In this paper a new Network Exterior Point Simplex Algorithm (NEPSA) for the Minimum Cost Network Flow Problem (MCNFP) is analytically presented. NEPSA belongs to a special simplex type category and is a modification of the classical network simplex algorithm. The main idea of the algorithm is to compute two flows. One flow is basic but not always feasible and the other is feasible but not always basic. A complete proof of correctness for the proposed algorithm is also presented. Moreover, the computational behavior of NEPSA is shown by an empirical study carried out for randomly generated sparse instances created by the well known gridgen network problem generator.