Maximum Matchings In Bipartite Graphs Research Articles

Test Optimization in Memristor Crossbars Based on Path Selection

Memristors have recently shown significant promise in designing memory and logic subsystems. A 2D-crossbar architecture built with memristor arrays provides a convenient platform for storing multivalued memory states by utilizing the analog variation of current-induced resistance through these cells. The integration of CMOS components with non-CMOS memristor cells further enhances the scope of their applications to various complex system designs. However, present-day memristors by virtue of their inherent structure are prone to various manufacturing defects and sensitive to operational modalities. Existing techniques for testing memristor arrays are either ad hoc in nature or suited for application-specific designs with little concern for optimizing test time. In this work, we envisage a 2-D memristor crossbar as a network and identify certain paths that are suitable for fault sensitization. For full-size square and rectangular memristive crossbars, the proposed method optimizes test time using a path-based technique guided by maximum matching in bipartite graphs. An integer linear programming (ILP) formulation is then used to solve the problem for a general crossbar, either full or incomplete. Simulation results with LTspice demonstrate the effectiveness and superiority of the method to the prior art in terms of test time and fault coverage.

Articulated Multi-Instrument 2-D Pose Estimation Using Fully Convolutional Networks.

Instrument detection, pose estimation, and tracking in surgical videos are an important vision component for computer-assisted interventions. While significant advances have been made in recent years, articulation detection is still a major challenge. In this paper, we propose a deep neural network for articulated multi-instrument 2-D pose estimation, which is trained on detailed annotations of endoscopic and microscopic data sets. Our model is formed by a fully convolutional detection-regression network. Joints and associations between joint pairs in our instrument model are located by the detection subnetwork and are subsequently refined through a regression subnetwork. Based on the output from the model, the poses of the instruments are inferred using maximum bipartite graph matching. Our estimation framework is powered by deep learning techniques without any direct kinematic information from a robot. Our framework is tested on single-instrument RMIT data, and also on multi-instrument EndoVis and in vivo data with promising results. In addition, the data set annotations are publicly released along with our code and model.

An o(n<sup>2.5</sup>) Algorithm: For Maximum Matchings in General Graphs

This article extend the John E. Hopcroft and Richart M. Karp Algorithm (HK Algorithm) for maximum matchings in bipartite graphs to the non-bipartite case by providing a new approach to deal with the blossom in alternating paths in the process of searching for augmenting paths, which different from well-known “shrinking” way of Edmonds and makes the algorithm for maximum matchings in general graphs more simple.

Anti-Ramsey numbers for matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(Kn, H) was introduced by Erdős, Simonovits and Sós in 1973, which is defined to be the maximum number of colors in an edge coloring of the complete graph Kn without any rainbow H. Later, the anti-Ramsey numbers for several special graph classes in complete are determined. Moreover, researchers generalized the host graph Kn to other graphs, in particular, to complete bipartite graphs and regular bipartite graphs. Li and Xu (2009) [18] proved that: Let G be a k-regular bipartite graph with n vertices in each partite set, then AR(G,mK2)=k(m−2)+1 for all m ≥ 2, k ≥ 3 and n>3(m−1). In this paper, we consider the anti-Ramsey number for matchings in 3-regular bipartite graphs. By using the known result that the vertex cover equals the size of maximum matching in bipartite graphs, we prove that AR(G,mK2)=3(m−2)+1 for n>32(m−1) when G is a 3-regular bipartite graph with n vertices in each partite set.

Raw Trajectory Rectification via Scene-Free Splitting and Stitching

Trajectories carry rich motion cues and thus have been leveraged to many high-level computer vision tasks. Due to the easy implementation of simple trackers, most previous work on trajectory-based applications utilizes raw tracking outputs without explicitly considering tracking errors. Reliable trajectories are prerequisite for modeling and recognizing high-level behaviors. Therefore, this paper tackles such problems by rectifying raw trajectories, which aims to post-process existing trajectories. Our approach firstly splits them into short tracks, and then infers identity ambiguity to remove unqualified detection responses. At last, short tracks are stitched via maximum bipartite graph matching. This postprocessing is completely scene-free. Results of trajectory rectification and their benefits are both evaluated on two challenging datasets. Results demonstrate that rectified trajectories are conducive to high-level tasks and the proposed approach is also competitive with state-of-the-art multi-target tracking methods.

On Stable Matchings and Flows

We describe a flow model related to ordinary network flows the same way as stable matchings are related to maximum matchings in bipartite graphs. We prove that there always exists a stable flow and generalize the lattice structure of stable marriages to stable flows. Our main tool is a straightforward reduction of the stable flow problem to stable allocations. For the sake of completeness, we prove the results we need on stable allocations as an application of Tarski’s fixed point theorem.

Real-Time Path Planning for Coordinated Transport of Multiple Particles Using Optical Tweezers

Automated transport of multiple particles using optical tweezers requires real-time path planning to move them in coordination by avoiding collisions among themselves and with randomly moving obstacles. This paper develops a decoupled and prioritized path planning approach by sequentially applying a partially observable Markov decision process algorithm on every particle that needs to be transported. We use an iterative version of a maximum bipartite graph matching algorithm to assign given goal locations to such particles. We then employ a three-step method consisting of clustering, classification, and branch and bound optimization to determine the final collision-free paths. We demonstrate the effectiveness of the developed approach via experiments using silica beads in a holographic tweezers setup. We also discuss the applicability of our approach and challenges in manipulating biological cells indirectly by using the transported particles as grippers.

Perfect matchings via uniform sampling in regular bipartite graphs

In this article we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first nontrivial algorithm, with running time O ( mn ), dates back to König's work in 1916 (here m = nd is the number of edges in the graph, 2 n is the number of vertices, and d is the degree of each node). The currently most efficient algorithm takes time O(m) , and is due to Cole et al. [2001]. We improve this running time to O (min{ m , n 2.5 ln n / d }); this minimum can never be larger than O ( n 1.75 √ln n ). We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a d -regular bipartite graph independently with a probability p = O ( n ln n / d 2 ) then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential witnesses to nonexistence of a matching (after sampling) in any piece and cuts of comparable size in that same piece. Karger's sampling theorem [1994a, 1994b] for preserving cuts in a graph can now be adapted to prove our uniform sampling theorem for preserving perfect matchings. Using the O ( m √ n ) algorithm (due to Hopcroft and Karp [1973]) for finding maximum matchings in bipartite graphs on the sampled graph then yields the stated running time. We also provide an infinite family of instances to show that our uniform sampling result is tight up to polylogarithmic factors (in fact, up to ln 2 n ).

Stabilizing maximum matching in bipartite networks

A matching $${E_\mathcal{M}}$$of graph G = (V, E) is a subset of the edges E, such that no vertex in V is incident to more than one edge in $${E_\mathcal{M}}$$. The matching $${E_\mathcal{M}}$$is maximum if there is no matching in G with size strictly larger than the size of $${E_\mathcal{M}}$$. In this paper, we present a distributed stabilizing algorithm for finding maximum matching in bipartite graphs based on the stabilizing PIF algorithm of Cournier et al. (Proceedings of 21st IEEE international conference on distributed computing systems, 91–98, 2001). Since our algorithm is stabilizing, it does not require initialization and withstands transient faults. The complexity of the proposed algorithm is O(d × n) rounds, where d is the diameter of the communication network and n is the number of nodes in the network. The space complexity is O(log Δ + log d), where Δ is the largest degree of all the nodes in the communication network. In addition, an optimal version of the proposed algorithm finding maximum matching in linear time is also presented.

Number of maximum matchings of bipartite graphs with positive surplus

A bipartite graph G=( A, B) is said to have positive surplus (as viewed from A) if the number of neighbors of X is bigger than the size of X for any non-empty subset X of A. In this paper, two lower bounds on the number of maximum matchings in bipartite graphs with positive surplus are obtained. The enumeration problems for maximum matchings in special bipartite graphs are solved. From this, the number of perfect matchings of some elementary bipartite graphs is given.

A Competitive Strong Spanning Tree Algorithm for the Maximum Bipartite Matching Problem

The new characterization for maximum matching in bipartite graphs given by Balinski and González is based on “strong spanning trees” and is independent on the classical notion of augmenting path. However, the algorithm that they derived runs in $0( | V | | E | )$ time for bipartite graphs with $| V |$ nodes and $| E |$ edges, and so it is not competitive with the $0( \sqrt {| V |} | E | )$ algorithm of Hopcroft and Karp. In this paper we prove that the basic results given by Hopcroft and Karp can also be obtained using the new characterization, allowing us to develop a competitive algorithm.

The Marriage Lemma for Polygamists

It has long been the ambition of one of the authors (it is left as an exercise to the reader to determine which one) to appear as a guest on one of the sensational TV talk shows. But what could a mathematician talk about that would be of interest to Geraldo or Donahue? The first subject that comes to mind is the so-called marriage problem, alias the assignment problem; here the objective is to have a matching between a set of women and a set of men given that each woman likes some of the men and dislikes the rest. Let the integer m represent the number of women; the number of men must, of course, be at least m. An obvious necessary condition that each woman gets herself a man is that for each p < m, every set of p women must together like at least p different men. Philip Hall's theorem, which has come to be known as the Marriage Lemma, asserts that this obvious necessary condition is sufficient. The proof of this result can most easily be accomplished by expressing the problem as that of determining the maximum flow in an appropriate network and then applying the maximum flow minimum cut theorem [9]. Another method of proof is based on the concept of maximum matchings in bipartite graphs that was developed by D. Konig and Philip Hall [1]. After further reflection, the authors suddenly had their moment of inspiration: Why not generalize the result to the polygamous case, i.e., where each man can be married to more than one woman? To be specific, for a fixed positive integer d ? 1 we assume that up to d women can be married to one man. In the monogamous case we clearly needed at least m men, but now the only obvious condition is that the number n of men must satisfy m < nd. Note that we do not require each man to marry d women; a complete matching requires only that all women are married. Note also that we do not require the compatibility to be symmetric. A woman may be married to a man if she likes him, even if he does not like her.

Graph search algorithms and maximum bipartite matching algorithm on the hypercube network model

In this paper, we propose parallel algorithms for breadth-first search and depth-first search on the hypercube network model. In addition, a parallel algorithm based on the same model for finding maximum matching in bipartite graphs is proposed.

An $n^{5/2} $ Algorithm for Maximum Matchings in Bipartite Graphs

The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.