A genuinely polynomial primal simplex algorithm for the assignment problem

Abstract

We present a primal simplex algorithm that solves the assignment problem in 1 2n(n+3)-4 pivots. Starting with a problem of size 1, we sequentially solve problems of size 2,3,4,...,n. The algorithm utilizes degeneracy by working with strongly feasible trees and employs Dantzig's rule for entering edges for the subproblem. The number of nondegenerate simplex pivots is bounded by n-1. The number of consecutive degenerate simplex pivots is bounded by 1 2(n-2)(n+1). All three bounds are sharp. The algorithm can be implemented to run in O(n3) time for dense graphs. For sparse graphs, using state of the art data structures, it runs in O(n2 log n+nm) time, where the bipartite graph has 2n nodes and m edges. © 1993

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

“…Hence by Theorem k if (i) holds, and x* < xe = 2, x* > xe = ue k I~ if (ii) holds, where x* is some optimal solution of (1 Since doubling all costs in problem(2 + 1) does not affect the optimality of a 2c~ +1 for any arc solution to that problem, and since the difference Ac2 = Ce --e e A is either 1, -1, or 0, we can solve problem(2) very easily given an optimal solution to problem(2 + 1). In solving problem (2), after doubling all costs we do not change the cost coefficients of all arcs in the network from 2c~ + 1 to Ce at the same time; rather we effect this change sequentially, simultaneously changing only some of the coefficients corresponding to the arcs adjacent to a specified node. For this purpose, we introduce working coefficients de = 2c2 + 1, Ve e A, when we start to solve problem (2).…”

“…In solving problem (2), after doubling all costs we do not change the cost coefficients of all arcs in the network from 2c~ + 1 to Ce at the same time; rather we effect this change sequentially, simultaneously changing only some of the coefficients corresponding to the arcs adjacent to a specified node. For this purpose, we introduce working coefficients de = 2c2 + 1, Ve e A, when we start to solve problem (2). After each change of working coefficients de of some arcs e in F(u) or B(u) from 2e~ + 1 to % we solve the resulting problem.…”

Linear Assignment Problems and Extensions

“…Goldfarb [97] proposed a similar algorithm based on signatures which solves a sequence of subproblems of the given LSAP. As in the algorithm of Akgül [7] the subproblems are LSAPs with cost matrices being submatrices of C = (c ij ). The cost matrix of the k-th subproblem is defined by the first k rows and the first k columns of C. Balinski's algorithm is used to solve the k-th subproblem, i.e., to produce a tree corresponding to a dual feasible basis with signature (2, .…”

“…Akgül [7] designed a primal simplex algorithm which performs O(n 2 ) pivots and has an overall worst case complexity of O(n 3 ). Also this algorithm uses strongly feasible trees as basic solutions and Dantzig's pivot rule to choose the variable which enters the basis.…”

A polynomial time primal network simplex algorithm for minimum cost flows

“…There are polynomial time primal network simplex algorithms for (i) the assignment problem (see, for example, Ahuja and Orlin [1992], Akgul [1993], Hung [1983], Orlin [1985], Roohy-Laleh [1980]), and Sokkalingam, Sharma and Ahuja [1993]; (ii) the shortest path problem (see, for example, , Akgul [1985], Dial, Glover, Karney, and Klingman [1979], Goldfarb, Hao, and Kai [1990], Orlin [1985]), and Sokkalingam, Sharma and Ahuja [1993]; and (iii) the maximum flow problem (see, for example, Goldberg, Grigoriadis, and Tarjan [1991] and 1991] ). There are also polynomial time dual network simplex algorithms for the minimum cost flow problem (see, for example, Orlin [1984], Orlin, Plotkin and Tardos [1993], and Plotkin and Tardos [1990],).…”