Abstract—We propose a new methodology for reliably solving the correspondence problem between sparse sets of points of two or more images. This is a key step in most problems of computer vision and, so far, no general method exists to solve it. Our methodology is able to handle most of the commonly used assumptions in a unique formulation, independent of the domain of application and type of features. It performs correspondence and outlier rejection in a single step and achieves global optimality with feasible computation. Feature selection and correspondence are first formulated as an integer optimization problem. This is a blunt formulation, which considers the whole combinatorial space of possible point selections and correspondences. To find its global optimal solution, we build a concave objective function and relax the search domain into its convex-hull. The special structure of this extended problem assures its equivalence to the original one, but it can be optimally solved by efficient algorithms that avoid combinatorial search. This methodology can use any criterion provided it can be translated into cost functions with continuous second derivatives. Index Terms—Correspondence problem, linear and concave programming, sparse stereo. 1

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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published

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.

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.

Internet routers frequently use a crossbar switch to interconnect linecards. The crossbar switch is scheduled using an algorithm that picks a new crossbar configuration every cycle. Several scheduling algorithms have been shown to guarantee 100% throughput under a variety of traffic patterns. The first such algorithm was the maximum weight matching (MWM) algorithm in which the weight is the sum of the occupancies of the queues. We explore whether alternative weight functions, such as using the sum of the square of the occupancies, leads to stronger or weaker stability. The first result of this paper is that a broad class of weight functions give rise to strong stability, including the sum of the squares, the sum of the cubes and so on. A counter-intuitive result, indicating a limitation of the Lyapunov method, is that the sum of the square root of the occupancies is not included in this class, even though simulation suggests that the resulting average delay is lower than for the other functions. We also consider the simpler, , randomized scheduling algorithm (TASS) proposed by Tassiulas. We show similar results for different weight functions as for MWM. We finally show that TASS gives 100% throughput when the weights are noisy, or out-of-date.

We present a parallel priority queue that supports the following operations in constant time: parallel insertion of a sequence of elements ordered according to key, parallel decrease key for a sequence of elements ordered according to key, deletion of the minimum key element, as well as deletion of an arbitrary element. Our data structure is the first to support multi insertion and multi decrease key in constant time. The priority queue can be implemented on the EREW PRAM, and can perform any sequence of n operations in O(n) time and O(m log n) work, m being the total number of keys inserted and/or updated. A main application is a parallel implementation of Dijkstra's algorithm for the single-source shortest path problem, which runs in O(n) time and O(m log n) work on a CREW PRAM on graphs with n vertices and m edges. This is a logarithmic factor improvement in the running time compared with previous approaches.

We present experimental results for four bipartite matching algorithms on 11 classes of graphs. The algorithms are depth-first search (dfs), breadth-first search (bfs), the push-relabel algorithm [GT88b], and the algorithm by Alt, Blum, Mehlhorn, and Paul (abmp) [ABMP91]. dfs was thought to be a good choice for bipartite matching but our results show that, depending on the input graph, it can have very poor performance. bfs on the other hand has generally very good performance. The results also show that the abmp and push-relabel implementations are similar in performance, but abmp was faster in most cases. We did not find a clear-cut advantage of abmp over bfs or vice-versa, but both the abmp and push-relabel implementations have generally smaller growth rates than bfs, and should thus be preferred if very large problems are to be solved. For small problems bfs is the best choice. We also present experimental results from a parallel implementation of the push-relabel algorithm, show...

In this paper we use interior-point methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Our algorithm finds a maximum cardinality matching in a bipartite graph with n nodes and m edges in O( p m log 3 n) time on a CRCW PRAM. Our results extend to the weighted bipartite matching problem and to the zero-one minimum-cost flow problem, yielding O( p m log 2 n log nC) algorithms, where C ? 1 is an upper bound on the absolute value of the integral weights or costs in the two problems, respectively. Our results improve previous bounds on these problems and introduce interior-point methods to the context of parallel algorithm design. 1 Introduction In this paper we use interior-point methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Although Karp, Upfal, and Wigderson [6] have sho...

Abstract. Given two undirected trees T and P, the Subtree Homeomorphism Problem is to find whether T has a subtree t that can be transformed into P by removing entire subtrees, as well as repeatedly removing a degree-2 node and adding the edge joining its two neighbors. In this paper we extend the Subtree Homeomorphism Problem to a new optimization problem by enriching the subtree-comparison with node-to-node similarity scores. The new problem, denoted ALSH (Approximate Labelled Subtree Homeomorphism) is to compute the homeomorphic subtree of T which also maximizes the overall node-to-node resemblance. We describe an O(m 2 n / log m + mn log n) algorithm for solving ALSH on unordered, unrooted trees, where m and n are the number of vertices in P and T, respectively. We also give an O(mn) algorithm for rooted ordered trees. 1