The cost scaling push-relabel method has been shown to be efficient for solving minimum-cost flow problems. In this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment's special structure. The results show that the method is very promising for practical use.

This paper is dedicated to the memory of Lillian Barros Abstract. Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros (1998) and Barros and Riley (2001) developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.

We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the push-relabel method is most efficient in practice and to compare push-relabel algorithms with augmenting path algorithms. We have implemented and compared three push-relabel algorithms, three augmenting path algorithms (one of which is new), and one augment-relabel algorithm. The depth-first search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowest-level selection push-relabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.

We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

Abstract — Updating probabilistic belief matrices as new observations arrive, in the presence of noise, is a critical part of many algorithms for target tracking in sensor networks. These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper addresses the problem of updating belief matrices to satisfy sum constraints using scaling algorithms. We show that the convergence behavior of the Sinkhorn scaling process, used for scaling belief matrices, can vary dramatically depending on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algorithm based on the maximum-flow algorithm that determines whether a given matrix is exactly scalable, thus determining the convergence properties of the Sinkhorn scaling process. We prove that the Sinkhorn scaling process always provides a solution to the problem of minimizing the Kullback-Leibler distance of the physically feasible scaled matrix from the prior constraint-violating matrix, even when the matrices are not exactly scalable. We pose the scaling process as a linearly constrained convex optimization problem, and solve it using an interior-point method. We prove that even in cases in which the matrices are not exactly scalable, the problem can be solved to ɛ−optimality in strongly polynomial time, improving the best known bound for the problem of scaling arbitrary nonnegative rectangular matrices to prescribed row and column sums. I.

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A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm. 1

The robustness function of a matroid measures the maximum increase in the weight of its minimum weight bases that can be produced by increases of a given total cost on the weights of its elements. We present an algorithm for computing this function, that runs in strongly polynomial time for matroids in which independence can be tested in strongly polynomial time. We identify key properties of transversal, scheduling and partition matroids, and exploit them to design robustness algorithms that are more efficient than our general algorithm.

Abstract. This paper shows that existing definitions of costs associated with soft global constraints are not sufficient to deal with all the usual global constraints. We propose more expressive definitions: refined variablebased cost, object-based cost and graph properties based cost. For the first two ones we provide ad-hoc algorithms to compute the cost from a complete assignment of values to variables. A representative set of global constraints is investigated. Such algorithms are generally not straightforward and some of them are even NP-Hard. Then we present the major feature of the graph properties based cost: a systematic way for evaluating the cost with a polynomial complexity. 1

. In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team i has w i wins and g ij games left to play against team j.Ateamis eliminated if it cannot possibly finish the season in first place or tied for first place. The goal is to determine exactly which teams are eliminated. The problem is not as easy as many sports writers would have you believe, in part because the answer depends not only on the number of games won and left to play, but also on the schedule of remaining games. In the 1960's, Schwartz showed how to determine whether one particular team is eliminated using a maximum flow computation. This paper indicates that the problem is not as di#cult as many mathematicians would have you believe. For each team i,letg i denote the number of games remaining. We prove that there exists a value W # such that team i is eliminated if and only if w i + g i <W # . Using this surprising fact, we can determine all eliminated team...