Let @math and @math be sets of @math red and @math blue points in the plane, respectively, with @math . Let @math be a perfect matching between points from @math and @math , using @math straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment in @math with @math , which we also call the [ ] value, bn of @math . We aim to find a matching under given constraints that minimizes @math . Any such matching is called a [ ] bottleneck matching of @math .
A variant of the bichromatic case is the so-called bicolored (or multicolored, when there are arbitrary many colors) case, where only the points of the same color are allowed to be matched. Abu-Affash, Bhore and Carmi in @cite_12 examined bicolored matchings that minimize the number of crossings between edges matching different color sets. They presented an algorithm to compute a bottleneck matching of points in convex position among all matchings that have no crossings of this kind.
@cite_12
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Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given n red points and n blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless P = NP. We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points. Let (X ) be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ( |X| 2 ) points. We show the existence of a non-crossing properly colored geometric perfect matching on (X ) (if (|X| ) is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most (3 ) on (X ). Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points. This paper presents an approach to object discovery in a given unlabeled image set, based on mining repetitive spatial configurations of image contours. Contours that similarly deform from one image to another are viewed as collaborating, or, otherwise, conflicting. This is captured by a graph over all pairs of matching contours, whose maximum a posteriori multicoloring assignment is taken to represent the shapes of discovered objects. Multicoloring is conducted by our new Coordinate Ascent Swendsen-Wang cut (CASW). CASW uses the Metropolis-Hastings (MH) reversible jumps to probabilistically sample graph edges, and color nodes. CASW extends SW cut by introducing a regularization in the posterior of multicoloring assignments that prevents the MH jumps to arrive at trivial solutions. Also, CASW seeks to learn parameters of the posterior via maximizing a lower bound of the MH acceptance rate. This speeds up multicoloring iterations, and facilitates MH jumps from local minima. On benchmark datasets, we outperform all existing approaches to unsupervised object discovery. This paper presents a framework of layered graph matching for integrating graph partition and matching. The objective is to find an unknown number of corresponding graph structures in two images. We extract discriminative local primitives from both images and construct a candidacy graph whose vertices are matching candidates (i.e., a pair of primitives) and whose edges are either negative for mutual exclusion or positive for mutual consistence. Then we pose layered graph matching as a multicoloring problem on the candidacy graph and solve it using a composite cluster sampling algorithm. This algorithm assigns some vertices into a number of colors, each being a matched layer, and turns off all the remaining candidates. The algorithm iterates two steps: 1) Sampling the positive and negative edges probabilistically to form a composite cluster, which consists of a few mutually conflicting connected components (CCPs) in different colors and 2) assigning new colors to these CCPs with consistence and exclusion relations maintained, and the assignments are accepted by the Markov Chain Monte Carlo (MCMC) mechanism to preserve detailed balance. This framework demonstrates state-of-the-art performance on several applications, such as multi-object matching with large motion, shape matching and retrieval, and object localization in cluttered background.