Perfect Matchings In Bipartite Graphs Research Articles

Bubble tracking method based on Kuhn-Munkres algorithm for boiling two-phase flow study

The study of bubble dynamics is important in various engineering applications as it can provide a fundamental understanding of the bubbly flow behavior. This paper presents a novel method for tracking bubble trajectories and classifying bubble behaviors, including formation, extinction, continuous movement, fragmentation, and coalescence. The method is based on the theory of perfect matching in bipartite graphs. By establishing the perfect matching between bubbles identified at adjacent time instants, judging the continuity of bubble motion, and simply calculating the number of bubbles in the neighbor of matching bubbles at adjacent time instants, the method can deal with the challenging situation where the bubble position and volume significantly change between two adjacent time instants. The method is validated in a simulated vertical pipe boiling flow, showing that the tracking accuracy is 100% and classification accuracies for continuous movement, fragmentation, and coalescence events are estimated to be larger than 90% in a wide range of time steps between two adjacent time instants.

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1±ϵ)-approximation for the number of perfect matchings of a λ-dense bipartite graph, using O(n1−2λλϵ−2) samples. With size n on each side and for 1 2>λ>0, a λ-dense bipartite graph has all degrees greater than (λ+1 2)n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

Randomized sequential importance sampling for estimating the number of perfect matchings in bipartite graphs

We introduce and study randomized sequential importance sampling algorithms for estimating the number of perfect matchings in bipartite graphs. In analyzing their performance, we establish various non-standard central limit theorems. We expect our methods to be useful for other applied problems.

Scaling matrices and counting the perfect matchings in graphs

We investigate efficient randomized methods for approximating the number of perfect matchings in bipartite graphs and general undirected graphs. Our approach is based on assigning probabilities to edges, randomly selecting an edge to be in a perfect matching, and discarding edges that cannot be put in a perfect matching. The probabilities are set according to the entries in the doubly stochastically scaled version of the adjacency matrix of the given graph. The experimental analysis on random and real-life graphs shows improvements in the approximation over previous and similar methods from the literature.

A deterministic parallel algorithm for bipartite perfect matching

A fundamental quest in the theory of computing is to understand the power of randomness. It is not known whether every problem with an efficient randomized algorithm also has one that does not use randomness. One of the extensively studied problems under this theme is that of perfect matching. The perfect matching problem has a randomized parallel (NC) algorithm based on the Isolation Lemma of Mulmuley, Vazirani, and Vazirani. It is a long-standing open question whether this algorithm can be derandomized. In this article, we give an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs. This gives us a deterministic parallel (quasi-NC) algorithm for the bipartite perfect matching problem. Derandomization of the Isolation Lemma means that we deterministically construct a weight assignment so that the minimum weight perfect matching is unique. We present three different ways of doing this construction with a common main idea.

Towards Boij–Söderberg theory for Grassmannians : the case of square matrices

We characterize the cone of GL-equivariant Betti tables of Cohen-Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for `Boij-S\"oderberg theory for Grassmannians', with the goal of characterizing the cones of GL_k-equivariant Betti tables of modules over the coordinate ring of k x n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k, C^n). The proof uses Hall's Theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fl{\o}ystad-Weyman.

On Directed 2-factors in Digraphs and 2-factors Containing Perfect Matchings in Bipartite Graphs

In this paper, we give the following result: If $D$ is a digraph of order $n$, and if $d_{D}^{+}(u) + d_{D}^{-}(v) \ge n$ for every two distinct vertices $u$ and $v$ with $(u, v) \notin A(D)$, then $D$ has a directed 2-factor with exactly $k$ directed cycles of length at least 3, where $n \ge 12k+3$. This result is equivalent to the following result: If $G$ is a balanced bipartite graph of order 2n with partite sets $X$ and $Y$, and if $d_{G}(x)+d_{G}(y) \ge n + 2$ for every two vertices $x \in X$ and $y \in Y$ with $xy \notin E(G)$, then for every perfect matching $M$, $G$ has a 2-factor with exactly $k$ cycles of length at least 6 containing every edge of $M$, where $n \ge 12k+3$. These results are generalizations of theorems concerning Hamilton cycles due to Woodall [Proc. Lond. Math. Soc., 24 (1972), pp. 739--755] and Las Vergnas [Problemes de couplages et problemes hamiltoniens en theorie des graphes, Ph.D. thesis, University of Paris, 1972], respectively.

Finding Perfect Matchings in Bipartite Hypergraphs

Haxell's condition [16] is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied.We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell's condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g. Set Packing.

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The perfect matching problem has a randomized NC-algorithm based on the Isolation Lemma of Mulmuley, Vazirani and Vazirani. We give an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs. This gives us a deterministic quasi-NC-algorithm for the bipartite perfect matching problem. The outline presented here emphasizes a geometric point of view. We think that this will be useful also for the perfect matching problem in general graphs.

Stochastic Enumeration Method for Counting Trees

The problem of estimating the size of a backtrack tree is an important but hard problem in the computational sciences. An efficient solution of this problem can have a major impact on the hierarchy of complexity classes. The first randomized procedure, which repeatedly generates random paths through the tree, was introduced by Knuth. Unfortunately, as was noted by Knuth and a few other researchers, the estimator can introduce a large variance and become ineffective in the sense that it underestimates the cost of the tree. Recently, a new sequential algorithm called Stochastic Enumeration (SE) method was proposed by Rubinstein et al. The authors showed numerically that this simple algorithm can be very efficient for handling different counting problems, such as counting the number of satisfiability assignments and enumerating the number of perfect matchings in bipartite graphs. In this paper we introduce a rigorous analysis of SE and show that it results in significant variance reduction as compared to Knuth’s estimator. Moreover, we establish that for almost all random trees the SE algorithm is a fully polynomial time randomized approximation scheme (FPRAS) for the estimation of the overall tree size.

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)).

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.

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.

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2 k , for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2 k , we also exhibit a highly parallel $$\oplus {\bf L}$$ algorithm for finding the value of a permanent modulo 2 k . Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2 k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.

A note on the hardness results for the labeled perfect matching problems in bipartite graphs

In this note, we strengthen the inapproximation bound of O(logn) for the labeled perfect matching problem established in J. Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters 96 (2005) 81-88, using a self improving operation in some hard instances. It is interesting to note that this self improving operation does not work for all instances. Moreover, based on this approach we deduce that the problem does not admit constant approximation algorithms for connected planar cubic bipartite graphs.

Perfect matchings and extended polymatroid

The classical conditions of the existence of a perfect matching in bipartite graphs are used directly or indirectly to solve the assignment problem by the well-known algorithms. In the paper, we define a vector and an extended polymatroid for a bipartite graph so that the bipartite graph has a perfect matching if and only if the vector is the basis of the extended polymatroid.

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti-vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d-factors in bipartite graphs, and perfect 2-matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006

The structure of stable marriage with indifference

We consider the stable marriage problem where participants are permitted to express indifference in their preference lists (i.e., each list can be partially ordered). We prove that, in an instance where indifference takes the form of ties, the set of strongly stable matchings forms a distributive lattice. However, we show that this lattice structure may be absent if indifference is in the form of arbitrary partial orders. Also, for a given stable marriage instance with ties, we characterise strongly stable matchings in terms of perfect matchings in bipartite graphs. Finally, we briefly outline an alternative proof of the known result that, in a stable marriage instance with indifference in the form of arbitrary partial orders, the set of super-stable matchings forms a distributive lattice.

Perfect Matching in General vs. Cubic Graphs: A Note on the Planar and Bipartite Cases

It is known that finding a perfect matching in a general graph is AC0-equivalent to finding a perfect matching in a 3-regular (i.e. cubic) graph. In this paper we extend this result to both, planar and bipartite cases. In particular we prove that the construction problem for perfect matchings in planar graphs is as difficult as in the case of planar cubic graphs like it is known to be the case for the famous Map Four-Coloring problem. Moreover we prove that the existence and construction problems for perfect matchings in bipartite graphs are as difficult as the existence and construction problems for a weighted perfect matching of O(m) weight in a cubic bipartite graph.

Finding all the perfect matchings in bipartite graphs

This paper describes an algorithm for finding all the perfect matchings in a bipartite graph. By using the binary partitioning method, our algorithm requires O( c( n+ m)+ n 2.5) computational effort and O( nm) memory storage, (where n denotes the number of vertices, m denotes the number of edges, and c denotes the number of perfect matchings in the given bipartite graph).