Matching In General Graphs Research Articles

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.

An efficient distributed algorithm for maximum matching in general graphs

We present a distributed algorithm for maximum cardinality matching in general graphs. On a general graph withn vertices, our algorithm requiresO(n 5/2) messages in the worst case. On trees, our algorithm computes a maximum matching usingO(n) messages after the election of a leader.

Maximum matchings in general graphs through randomization

A new randomized algorithm for the maximum matching problem is presented. Unlike conventional matching algorithms which are combinatorial, our algorithm is algebraic and works on the Tutte matrix of the given graph. Although slower than the best known matching algorithm, our algorithm has the advantage of being conceptually simple and easy to program.

Efficient implementation of graph algorithms using contraction

The ( component ) merging problem is a new graph problem. Versions of this problem appear as bottlenecks in various graph algorithms. A new data structure solves this problem efficiently, and two special cases of the problem have even more efficient solutions based on other data structures. The performance of the data structures is sped up by introducing a new algorithmic tool called packets . The algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures. Specifically, Β-trees can be used simultaneously as a priority queue and a concatenable queue. Similarly, F-heaps support some kinds of split operations with no loss of efficiency. An immediate application of the solution to the simplest version of the merging problem is an Ο( t ( m , n )) algorithm for finding minimum spanning trees in undirected graphs without using F-heaps, where t ( m , n ) = m log 2 log 2 log d n , the graph has n vertices and m edges, and d = max( m / n , 2). Packets also improve the F-heap minimum spanning tree algorithm, giving the fastest algorithm currently known for this problem. The efficient solutions to the merging problem and the new observation about F-heaps lead to an Ο( n ( t ( m , n ) + n log n )) algorithm for finding a maximum weighted matching in general graphs. This settles an open problem posed by Tarjan [ 15, p. 123], where the weaker bound of O ( nm log ( n 2 / m )) was conjectured.

Sequential and Parallel Algorithms for Finding Maximum Matchings in Graphs

Graph algorithms use various techniques to achieve higher efficiency. In this review we exemplify the design and analysis of efficient graph algorithms by surveying algorithms for the four closely related problems of finding a maximum cardinality or weighted matching in general or bipartite graphs. Mathematicall y, these are all special cases of the problem of weighted matching in general graphs. However, we consider them in increasing order of difficulty because the easier the problem , the faster or simpler its solution. First we discuss sequential algorithms, and then we describe parallel algo­ rithms. Surprisingl y, at the moment, entirely different techniques are used for sequential and parallel algorithms. For sequential algorithms our computational model is the uniform cost random access machine or RAM (Aho et al 1974). Given a problem , we first try to solve it in polynomial time . Once a polynomial-time solution is found, we try to improve the solution by speeding it up. Our parallel computational model is the PRAM (Fortune & Wyllie 1978). It is a parallel computer with p processors (RAMs) that share a common memory. We allow only simultaneous reads from , but not simultaneous writes to, the same memory location. Given a problem, we first try to show that it is in the class NC (Cook 1983) by designing an algorithm that uses a polynomial number of processors and poly-logarithmic time. We call such an algorithm an NC algorithm. Alternative ly, if we use randomizat ion, we design an RNC algorithm. Once we establish that the problem is in NC or

An $O(EV\log V)$ Algorithm for Finding a Maximal Weighted Matching in General Graphs

We define two generalized types of a priority queue by allowing some forms of changing the priorities of the elements in the queue. We show that they can be implemented efficiently. Consequently, each operation takes $O(\log n)$ time. We use these generalized priority queues to construct an $O(EV\log V)$ algorithm for finding a maximal weighted matching in general graphs.

Motion detection and analysis of matching graphs of intermediate-level primitives

Introduces a new primitive, the half-chunk, for encoding boundary, region, and surface information in graph form. General graph matching is discussed and a specific graph matching system based on half-chunk graphs is presented. Following matching, bindings are interpreted as object component motion or depth. The half-chunk graph allows scale and coordinate system independent object models to be represented. Half-chunk graphs may encode 2-D or 3-D structure equally well.

The Nucleolus of a Characteristic Function Game

Abstract : In RM 23, a proof was given that the nucleolus is continuous as a function of the characteristic function. This proof is not correct; the author, at least, does not know how to complete it. In the paper a correct proof for this fact is given. The proof is based on an alternative definition of the nucleolus, which is of some interest in its own right. (Author)