Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

Recognition of d-dimensional Monge arrays

AbstractIt is known that the d-dimensional axial transportation (assignment) problem can easily be solved by a greedy algorithm if and only if the underlying cost array fulfills the d-dimensional Monge property. In this paper the following question is solved: Is it possible to find d permutations in such a way that the permuted array becomes a Monge array? Furthermore we give an algorithm which constructs such permutations in the affirmative case. If the cost array has the dimensions n1×n2×⋯×nd with n1⩽n2⩽⋯⩽nd, then the algorithm has time complexity O(d2n2⋯nd(n1+lognd)). By using this algorithm a wider class of d-dimensional axial transportation problems and in particular of the d-dimensional axial assignment problems can be solved efficiently

On Monge sequences in d-dimensional arrays

AbstractLet C be an n × m matrix. Then the sequence j:= ((i1, j1), (i2, j2), …, (inm, jnm)) of pairs of indices is called a Monge sequence with respect to the given matrix C if and only if, whenever (i, j) precedes both (i, s) and (r, j) in j, then c[i, j] + c[r, s] ≤ c[i, s] + c[r, j]. Monge sequences play an important role in greedily solvable transportation problems. Hoffman showed that the greedy algorithm which maximizes all variables along a sequence j in turn solves the classical Hitchcock transportation problem for all supply and demand vectors if and only if j is a Monge sequence with respect to the cost matrix C. In this paper we generalize Hoffman's approach to higher dimensions. We first introduce the concept of a d-dimensional Monge sequence. Then we show that the d-dimensional axial transportation problem is solved to optimality for arbitrary right-hand sides if and only if the sequence j applied in the greedy algorithm is a d-dimensional Monge sequence. Finally we present an algorithm for obtaining a d-dimensional Monge sequence which runs in polynomial time for fixed d