Bipartite Graph Matching Research Articles

Large matchings in bipartite graphs have a rainbow matching

Let g(n) be the least number such that every collection of n matchings, each of size at least g(n), in a bipartite graph, has a full rainbow matching. Aharoni and Berger (2009) conjectured that g(n)=n+1 for every n>1. This generalizes famous conjectures of Ryser, Brualdi and Stein. Recently, Aharoni et al. proved that g(n)≤⌊74n⌋. We prove that g(n)≤⌊53n⌋.

Uncertainty-Aware Sensor Deployment Strategy in Mixed Wireless Sensor Networks

Deployment is a fundamental issue in wireless sensor networks (WSNs), which affects the performance and lifetime of the networks. Usually the sensor locations are precomputed based on “perfect” sensor detection model, whereas sensors may not always provide reliable information, either due to operational tolerance levels or environmental factors. Therefore, it is very important to take into account this uncertainty in the deployment process. In this paper, we address the problem of sensor deployment in a mixed sensor network where the mobile and static nodes work collaboratively to perform deployment optimization task. We consider the Gaussian white noise in the environment and present a centralized algorithm (FABGM for short) which discoveres vacancies by using detection model based on false alarm and moves the mobile nodes according to the method based on bipartite graph matching in this study. In this algorithm, the management node of the WSNs collects the geographical information of all of the static and mobile sensors. Then, the management node executes the algorithm to get the best matches between mobile sensors and coverage holes. Simulation results are presented to demonstrate the effectiveness of the proposed approach.

TOPICVIEW: VISUAL ANALYSIS OF TOPIC MODELS AND THEIR IMPACT ON DOCUMENT CLUSTERING

We present a new approach for analyzing topic models using visual analytics. We have developed TopicView, an application for visually comparing and exploring multiple models of text corpora, as a prototype for this type of analysis tool. TopicView uses multiple linked views to visually analyze conceptual and topical content, document relationships identified by models, and the impact of models on the results of document clustering. As case studies, we examine models created using two standard approaches: Latent Semantic Analysis (LSA) and Latent Dirichlet Allocation (LDA). Conceptual content is compared through the combination of (i) a bipartite graph matching LSA concepts with LDA topics based on the cosine similarities of model factors and (ii) a table containing the terms for each LSA concept and LDA topic listed in decreasing order of importance. Document relationships are examined through the combination of (i) side-by-side document similarity graphs, (ii) a table listing the weights for each document's contribution to each concept/topic, and (iii) a full text reader for documents selected in either of the graphs or the table. The impact of LSA and LDA models on document clustering applications is explored through similar means, using proximities between documents and cluster exemplars for graph layout edge weighting and table entries. We demonstrate the utility of TopicView's visual approach to model assessment by comparing LSA and LDA models of several example corpora.

A Two-Stage Decomposition of High School Timetabling applied to cases in Denmark

Integer Programming (IP) has been used to model educational timetabling problems since the very early days of Operations Research. It is well recognized that these IP models in general are hard to solve, and this area of research is dominated by heuristic solution approaches. In this paper a Two-Stage Decomposition of an IP model for a practical case of high school timetabling is shown. This particular timetabling problem consists of assigning lectures to both a timeslot and a classroom, which is modeled using a very large amount of binary variables. The decomposition splits this model into two separate problems (Stage I and Stage II) with far less variables. These two separate problems are solved in sequence, such that the solution for the Stage I model is given as input to the Stage II model, implying that irreversible decisions are made in Stage I. However, the objective of the Stage II model is partly incorporated in the Stage I model by exploiting that Stage II can be seen as a minimum weight maximum matching problem in a bipartite graph. This theoretically strengthens the decomposition in terms of global optimality. The approach relies on Hall's theorem for the existence of matchings in bipartite graphs, which in its basic form yields an exponential amount of constraints in the Stage I model. However, it is shown that only a small subset of these constraints is needed, making the decomposition tractable in practice for IP solvers. To evaluate the decomposition, 100 real-life problem instances from the database of the high school ERP system Lectio are used. Computational results show that the decomposition performs significantly better than solving the original IP, in terms of both found solutions and bounds.

On a matching distance between rooted phylogenetic trees

Abstract The Robinson-Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values of similarity between clusters are transformed to the final MC-score of the dissimilarity of trees. The analyzed properties give insight into the structure of the metric space generated by MC, its relations with the Matching Split (MS) distance of unrooted trees and asymptotic behavior of the expected distance between binary n-leaf trees selected uniformly in both MC and MS (Θ(n3/2)).

Edge-coloring almost bipartite multigraphs

Bipartite multigraphs have chromatic index equal to the largest degree d. We consider multigraphs obtained by inserting k vertices in edges of a connected bipartite multigraph, and show that the chromatic index may increase to at most d+1. We further show that (1) the chromatic index always increases if k=1 when the original bipartite multigraph is regular, but does not increase for k=1 if the bipartite multigraph is not regular; (2) the chromatic index does not increase if k=2 (regular or not); (3) the chromatic index may increase if k=3 when the original bipartite multigraph is regular, but does not increase for k=3 if the bipartite multigraph is not regular; (4) the chromatic index does not increase if k=4 (regular or not); (5) the chromatic index increases for infinitely many 3-regular bipartite graphs and each k⩾3 odd, and (6) the chromatic index does not increase for infinitely many 3-regular bipartite graphs and each k⩾3 (even or odd).We finally study the list chromatic index of such multigraphs and show that it does not increase past d+1 either, in fact in some sense it stays almost d. All the results presented here render polynomial-time algorithms, since the proofs are based on bipartite graph matching and stable marriage.

Matchings in Random Biregular Bipartite Graphs

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdös and Rényi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.

New results on maximum induced matchings in bipartite graphs and beyond

The maximum induced matching problem is known to be APX-hard in the class of bipartite graphs. Moreover, the problem is also intractable in this class from a parameterized point of view, i.e. it is W[1]-hard. In this paper, we reveal several classes of bipartite (and more general) graphs for which the problem admits fixed-parameter tractable algorithms. We also study the computational complexity of the problem for regular bipartite graphs and prove that the problem remains APX-hard even under this restriction. On the other hand, we show that for hypercubes (a proper subclass of regular bipartite graphs) the problem admits a simple solution.

Detecting signage and doors for blind navigation and wayfinding

Signage plays a very important role to find destinations in applications of navigation and wayfinding. In this paper, we propose a novel framework to detect doors and signage to help blind people accessing unfamiliar indoor environments. In order to eliminate the interference information and improve the accuracy of signage detection, we first extract the attended areas by using a saliency map. Then the signage is detected in the attended areas by using a bipartite graph matching. The proposed method can handle multiple signage detection. Furthermore, in order to provide more information for blind users to access the area associated with the detected signage, we develop a robust method to detect doors based on a geometric door frame model which is independent to door appearances. Experimental results on our collected datasets of indoor signage and doors demonstrate the effectiveness and efficiency of our proposed method.

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.

SUBOPTIMAL GRAPH ISOMORPHISM USING BIPARTITE MATCHING

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.

Model-based recognition of 2D objects under perspective distortion

We report on a case study showing on recognition of objects under perspective distortion in projected 2d images. We use symbolic descriptions and yield similar results as heuristic or statistical methods. The knowledge is modeled in so-called TGraphs which are typed, attributed, and ordered directed graphs. We combine the search in the state space with a maximum weight bipartite graph-matching and in consequence we reduce the numerous amount of hypotheses. Furthermore we use hash tables to increase the runtime efficiency. As a result we reduce the runtime up to a factor of five in comparison to the system without hash tables and achieve a detection rate of 90.6% for a data set containing 968 perspective images of poker cards and domino tiles. Therefore, we show that model-based object recognition using symbolic descriptions is on a competitive basis.

Decomposition of binary images—A survey and comparison

We present an overview of the most important methods that decompose an arbitrary binary object into a union of rectangles. We describe a run-length encoding and its generalization, decompositions based on quadtrees, on mathematical morphology, on the distance transform, and a theoretically optimal decomposition based on a maximal matching in bipartite graphs. We compare their performance in image compression, in moment computation and in linear filtering. We show that the choice is always a compromise between the complexity and time/memory consumption. We give advice how to select an appropriate method in particular cases.

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n and |R| = m. We construct a random graph G by allowing each x∈L to choose d random neighbours in R. The question discussed is as to the size μ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is μ(G) = n whp? We answer this question exactly when d is at least three. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Induced Matchings in Subcubic Planar Graphs

Induced Matchings in Subcubic Planar Graphs

Rainbow Matchings in Properly Colored Bipartite Graphs

Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition (X,Y) and . We show that if , then G has a rainbow coloring of size at least .

Mining User Movement Behavior Patterns in a Mobile Service Environment

Mobile service systems offer users useful information ubiquitously via mobile devices. Based on changeable user movement behavior patterns (UMBPs), mobile service systems have the capability of effectively mining a special request from abundant data. In this paper, UMBPs are studied in terms of the problem of mining matching mobile access patterns based on joining the following four kinds of characteristics, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> , and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> , where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> is the mobile user, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> is the movement location, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> is the dwell time in the timestamp, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> is the service request. By introducing standard graph-matching algorithms along with the primitives of a database management system, which comprises grouping, sorting, and joining, these joint operations are defined. Moreover, by mining the associated structure via maximum weight bipartite graph matching, a prediction mechanism, based on the model of UMBPs, is utilized to find strong relationships among <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> , and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> . In addition, a PC-based experimental evaluation under various simulation conditions, using synthetically generated data, is introduced. Finally, performance studies are conducted to show that, in terms of execution efficiency and scalability, the proposed procedures produced excellent performance results.

A generalization of Hungarian method and Hall’s theorem with applications in wireless sensor networks

In this paper, we consider various problems concerning quasi-matchings and semi-matchings in bipartite graphs, which generalize the classical problem of determining a perfect matching in bipartite graphs. We prove a generalization of Hall’s marriage theorem, and present an algorithm that solves the problem of determining a lexicographically minimum g-quasi-matching (that is a set F of edges in a bipartite graph such that in one set of the bipartition every vertex v has at least g(v) incident edges from F, where g is a so-called need mapping, while on the other side of the bipartition the distribution of degrees with respect to F is lexicographically minimum). We obtain that finding a lexicographically minimum quasi-matching is equivalent to minimizing any strictly convex function on the degrees of the A-side of a quasi-matching and use this fact to prove a more general statement: the optima of any component-based strictly convex cost function on any subset of L1-sphere in Nn are precisely the lexicographically minimal elements of this subset. We also present an application in designing optimal CDMA-based wireless sensor networks.

An Interactive Map Search System Using Wavelet and Shape Contexts

In this paper a new interactive map search system is presented using shape context and bipartite graph matching. Shape context is used for measuring shape similarity and the recovering of point correspondences. After the above information is generated from the shape context bipartite graph matching is used to obtain the optimal correspondence between two shapes. Hierarchical description is also used to increase the recognition rate. Shape context is a method to treat shapes as a set of points and generate the histogram of the distribution of points. Wavelet analysis is used in hierarchical description. In order to shorten the calculation time, piecewise linear approximation is implemented as the feature extraction method. The systemlists the sixmost similar shapes to hand-written input shapes from the reference shapes, i.e., Japan’s 47 prefectures. Comparison results of linear matching, Dynamic Programming (DP) matching and shape context with bipartite graph matching indicate that the 1st place recognition rates are 82%, 84.52% and 92.45%, respectively. The evaluation result of hierarchical description shows that hierarchical approximation can improve the recognition rate from 92.45 to 94.97% using the deepest-4 depth. These results show that the proposed method is effective on fulfilling the interactive map search system.

Space complexity of perfect matching in bounded genus bipartite graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the log-space complexity class Stoic Probabilistic Log-space (SPL). Since SPL is contained in the log-space counting classes ⊕L (in fact in ModkL for all k⩾2), C=L, and PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs can be performed by a log-space transducer with an SPL oracle. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a log-space computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.