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Robotic Process Automation has emerged in recent years as an important field by allowing faster and more secure processes through a reduction in the risks or errors but also an increase in the productivity rates of many industries. In this specific paper, the RPA problem aims at assigning financial transactions to software robots to minimize the total costs, induced by the licenses and utilization time. The problem is represented as a bipartite graph and the effective resistance of the graph, which is analog to an electrical circuit, is used to order the edges of a heuristic method to assign the transactions to robots. Preliminary results, based on real data from a bank, are compared to the optimal solution obtained by a linear integer programming model. They show that the heuristic method allows to obtain results quicker and that they are near the optimal solution. [ABSTRACT FROM AUTHOR]

Finding a maximum-weight matching is a classical and well-studied problem in computer science, solvable in cubic time in general graphs. We introduce and consider in this work the "discovery" variant of the bipartite matching problem (or assignment problem) where edge weights are not provided as input but must be queried, requiring additional and costly computations. Hence, discovery algorithms are developed aiming to minimize the number of queried weights while providing guarantees on the computed solution. We show in this work the hardness of the underlying problem in general while providing several efficient algorithms that can make use of natural assumptions about the order in which the nodes are processed by the greedy algorithms. Our motivations for exploring this problem stem from finding practical solutions to maximum-weight matching in hypergraphs, a problem recently emerging in the formation of peer-to-peer energy sharing communities. [ABSTRACT FROM AUTHOR]

The Assignment Problem is a basic combinatorial optimization problem. In a weighted bipartite graph, the Assignment Problem is to find the largest sum of weights matching. The Hungarian method is a well-known algorithm, which is combinatorial optimization. Adding a new row and a new column to a weighted bipartite graph is called the Incremental Assignment Problem (IAP). The algorithm of the Incremental Assignment Problem utilizes the given optimal solution (the maximum weighted matching) and the dual variables to solve the matrix after extending the bipartite graph. This paper proposes an improvement of the Incremental Assignment Algorithm (IAA), named the Improved Incremental Assignment Algorithm (IIAA). The improved algorithm will save the operation time and operation space to find the optimal solution (the maximum weighted matching) of the bipartite graph. [ABSTRACT FROM AUTHOR]

We consider combinatorial algorithms for the minimum-cost flow problem on networks with unit capacities, and special cases of the problem. Historically, researchers have developed special-purpose algorithms that exploit unit capacities. In contrast, for the maximum flow problem, the classical blocking flow and push-relabel algorithms for the general case also have the best bounds known for the special case of unit capacities. In this paper we show that the classical blocking flow push-relabel cost-scaling algorithms of Goldberg and Tarjan (Math. Oper. Res. 15, 430-466, 1990) for general minimum-cost flow problems achieve the best known bounds for unit-capacity problems as well. We also develop a cycle-canceling algorithm that extends Goldberg's shortest path algorithm (Goldberg SIAM J. Comput. 24, 494-504, 1995) to minimum-cost, unit-capacity flow problems. Finally, we combine our ideas to obtain an algorithm that solves the minimum-cost bipartite matching problem in $O(r^{1/2} m \log C)$ time, where m is the number of edges, C is the largest arc cost (assumed to be greater than 1), and r is the number of vertices on the small side of the vertex bipartition. This result generalizes (and simplifies) a result of Duan et al. (2011) and solves an open problem of Ramshaw and Tarjan (2012). [ABSTRACT FROM AUTHOR]

The bipartite record linkage task consists of merging two disparate datafiles containing information on two overlapping sets of entities. This is nontrivial in the absence of unique identifiers and it is important for a wide variety of applications given that it needs to be solved whenever we have to combine information from different sources. Most statistical techniques currently used for record linkage are derived from a seminal article by Fellegi and Sunter in 1969. These techniques usually assume independence in the matching statuses of record pairs to derive estimation procedures and optimal point estimators. We argue that this independence assumption is unreasonable and instead target a bipartite matching between the two datafiles as our parameter of interest. Bayesian implementations allow us to quantify uncertainty on the matching decisions and derive a variety of point estimators using different loss functions. We propose partial Bayes estimates that allow uncertain parts of the bipartite matching to be left unresolved. We evaluate our approach to record linkage using a variety of challenging scenarios and show that it outperforms the traditional methodology. We illustrate the advantages of our methods merging two datafiles on casualties from the civil war of El Salvador. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The well-known problem of weighted matching in an arbitrary graph H with n vertices is reduced to a matching problem for a bipartite graph with 2n vertices. The maximum matching of graph H with the minimum sum of weights of edges defined by matrix[c] is found in timeO ( n ) after ordering, in non-descending order, the values above the main diagonal. [ABSTRACT FROM AUTHOR]

The paper considers the minimization of the sum of weights of edges forming a subset U′ ⊂ U of the set of disjoint simple cycles at vertices υ ϵ V in graph H = (V,U) and covering V. The problem under study (2-f problem) is polynomially solvable in algorithms that are characterized by technical difficulties that hinder the acceleration of computing. The 2-f problem is solved by reducing it to a simpler bipartite case. The result is represented by a perfect matching of bipartite graph corresponding to the solution of the assignment problem, where each circuit of cyclic expansion contains at least three arcs. [ABSTRACT FROM AUTHOR]

We consider the bilevel assignment problem in which each decision maker (i.e., the leader and the follower) has its own objective function and controls a distinct set of edges in a given bipartite graph. The leader acts first by choosing some of its edges. Subsequently, the follower completes the assignment process. The edges selected by the leader and the follower are required to constitute a perfect matching. In this paper we propose an exact solution approach, which is based on a branch-and-bound framework and exploits structural properties of the assignment problem. Extensive computational experiments with linear sum and linear bottleneck objective functions are conducted to demonstrate the performance of the developed methods. While the considered problem is known to be NP-hard in general, we also describe some restricted cases that can be solved in polynomial time. [ABSTRACT FROM AUTHOR]

The paper proposes a new method to solve the assignment problem based on recursive derivation of the optimal solution. The assignment problem is formulated in the rearrangement matrix form that allows the use of the matrix approach to optimal solution. The algorithm is to find the minimum total weight matching in the bipartite graph with 2n vertices. The computational scheme of the recurrence method for solving the assignment problem is presented in the form adapted for implementation on a computer. [ABSTRACT FROM AUTHOR]

Graphs provide us with a flexible and powerful way to represent objects in various areas of computer science. One of the main drawbacks is, however, that many standard algorithms on graphs have a high computational complexity. The present paper considers the problem of graph isomorphism, i.e. checking two graphs for identity. A novel approach for the efficient computation of graph isomorphism is presented. The proposed algorithm is based on bipartite graph matching by means of an assignment algorithm. The algorithmic framework is suboptimal in the sense of possibly rejecting pairs of graphs without making a decision. As an advantage, however, it offers polynomial runtime. In experiments on diverse graph data sets we demonstrate substantial speedups of our proposed method over several standard procedures for graph isomorphism. Furthermore, although the computational framework for isomorphism is suboptimal, we show that the proposed algorithm rejects only very few pairs of graphs and otherwise returns correct results. [ABSTRACT FROM AUTHOR]

We consider the problem of multi-robot task-allocation when robots have to deal with uncertain utility estimates. Typically an allocation is performed to maximize expected utility; we consider a means for measuring the robustness of a given optimal allocation when robots have some measure of the uncertainty (e.g. a probability distribution, or moments of such distributions). We introduce the interval Hungarian algorithm, a new algorithm that extends the classic Kuhn—Munkres Hungarian algorithm to compute the maximum interval of deviation, for each entry in the assignment matrix, which will retain the same optimal assignment. The algorithm has a worst-case time complexity of O(n4); we also introduce a parallel variant with O(n3) running time, which is able to exploit the concurrent computing capabilities of distributed multi-robot systems. This provides an efficient measurement of the tolerance of the allocation to the uncertainties and dynamics, for both a specific interval and a set of interrelated intervals. We conduct experiments both in simulation and with physical robots to validate the approach and to gain insight into the effect of location uncertainty on allocations for multi-robot multi-target navigation tasks. [ABSTRACT FROM AUTHOR]

Moving object databases (MOD) are being used in a wide range of location-based services that are of growing interest in many application areas. In the literature, several query types such as nearest neighbor, reverse nearest neighbor, k-nearest neighbor, and proximity queries have been considered in MOD. In this paper, we propose a novel operator called the assignment operator as a query type for MOD. The assignment operator is an operator used in a query to solve the assignment problem (also known as the weighted bipartite graph-matching problem). Assignment operator finds a perfect match between two sets of objects in a manner that minimizes a total cost. For instance, a set of moving objects such as taxi cabs are assigned to a set of customers in a manner that minimizes the total cost of traveling for the taxis. A possible implementation of the assignment operator in MOD and its performance evaluation are given. [ABSTRACT FROM AUTHOR]

We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems. [Copyright &y& Elsevier]

In K(n,n) with edges colored either red or blue, we show that the problem of finding a solution matching, a perfect matching consisting of exactly r red edges, and (n−r) blue edges for specified 0⩽r⩽n, is a nontrivial integer program. We present an alternative, logically simpler proof of a theorem in (Kibernetika 1 (1987) 7–11) which establishes necessary and sufficient conditions for the existance of a solution matching, and a new O(n2.5) algorithm. This shows that the problem of finding an assignment of specified cost r in an assignment problem on the complete bipartite graph with a 0−1 cost matrix is efficiently solvable. [Copyright &y& Elsevier]

Periodic global updates of dual variables have been shown to yield a substantial speed advantage in implementations of push-relabel algorithms for the maximum flow and minimum cost flow problems. In this paper, we show that in the context of the bipartite matching and assignment problems, global updates yield a theoretical improvement as well. For bipartite matching, a push-relabel algorithm that uses global updates runs in O(√nm &frac log(N² /m) / log n;time (matching the best bound known) and performs worse by a factor of √n without the updates. A similar result holds for the assignment problem, for which an algorithm that assumes integer costs in the range [- C, …, C] and that runs in time O(√nm log(nC)) (matching the best cost-scaling bound known) is presented. [ABSTRACT FROM AUTHOR]