Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

On the induced matching problem

We study extremal questions on induced matchings in certain natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of size at least n/40 while there are planar twinless graphs that do not contain an induced matching of size (n+10)/27. We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation. For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40k that is computable in linear time; this significantly improves the results of Moser and Sikdar (2007). We also show that we can decide in time O(k⁹¹+n) whether a planar graph contains an induced matching of size at least k.     

Uniquely Restricted Matchings

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs. Received April 12, 1998; revised June 21, 1999.     

eTuner: tuning schema matching software using synthetic scenarios

Most recent schema matching systems assemble multiple components, each employing a particular matching technique. The domain user mustthen tune the system: select the right component to be executed and correctly adjust their numerous “knobs” (e.g., thresholds, formula coefficients). Tuning is skill and time intensive, but (as we show) without it the matching accuracy is significantly inferior. We describe eTuner, an approach to automatically tune schema matching systems. Given a schema S, we match S against synthetic schemas, for which the ground truth mapping is known, and find a tuning that demonstrably improves the performance of matching S against real schemas. To efficiently search the huge space of tuning configurations, eTuner works sequentially, starting with tuning the lowest level components. To increase the applicability of eTuner, we develop methods to tune a broad range of matching components. While the tuning process is completely automatic, eTuner can also exploit user assistance (whenever available) to further improve the tuning quality. We employed eTuner to tune four recently developed matching systems on several real-world domains. The results show that eTuner produced tuned matching systems that achieve higher accuracy than using the systems with currently possible tuning methods.     

Feature-Based Sequence-to-Sequence Matching

This paper studies the problem of matching two unsynchronized video sequences of the same dynamic scene, recorded by different stationary uncalibrated video cameras. The matching is done both in time and in space, where the spatial matching can be modeled by a homography (for 2D scenarios) or by a fundamental matrix (for 3D scenarios). Our approach is based on matching space-time trajectories of moving objects, in contrast to matching interest points (e.g., corners), as done in regular feature-based image-to-image matching techniques. The sequences are matched in space and time by enforcing consistent matching of all points along corresponding space-time trajectories. By exploiting the dynamic properties of these space-time trajectories, we obtain sub-frame temporal correspondence (synchronization) between the two video sequences. Furthermore, using trajectories rather than feature-points significantly reduces the combinatorial complexity of the spatial point-matching problem when the search space is large. This benefit allows for matching information across sensors in situations which are extremely difficult when only image-to-image matching is used, including: (a) matching under large scale (zoom) differences, (b) very wide base-line matching, and (c) matching across different sensing modalities (e.g., IR and visible-light cameras). We show examples of recovering homographies and fundamental matrices under such conditions.     

Combinatorial and Experimental Methods for Approximate Point Pattern Matching

Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modeling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high-degree polynomials, and require robust calculations of intersection points of high-degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average-case analysis and a detailed empirical study of our methods.     

A Heuristic for Dijkstra''s Algorithm with Many Targets and Its Use in Weighted Matching Algorithms

Abstract. We consider the single-source many-targets shortest-path (SSMTSP) problem in directed graphs with non-negative edge weights. A source node s and a target set T is specified and the goal is to compute a shortest path from s to a node in T . Our interest in the shortest path problem with many targets stems from its use in weighted bipartite matching algorithms. A weighted bipartite matching in a graph with n nodes on each side reduces to n SSMTSP problems, where the number of targets varies between n and 1 . The SSMTSP problem can be solved by Dijkstra's algorithm. We describe a heuristic that leads to a significant improvement in running time for the weighted matching problem; in our experiments a speed-up by up to a factor of 12 was achieved. We also present a partial analysis that gives some theoretical support for our experimental findings.     

On the complexity of distributed stable matching with small messages

We consider the distributed complexity of the stable matching problem (a.k.a. “stable marriage”). In this problem, the communication graph is undirected and bipartite, and each node ranks its neighbors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if messages may contain at most B bits each, then any distributed algorithm that solves the stable matching problem requires ${\Omega(\sqrt{n/B\log n})}We consider the distributed complexity of the stable matching problem (a.k.a. “stable marriage”). In this problem, the communication graph is undirected and bipartite, and each node ranks its neighbors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if messages may contain at most B bits each, then any distributed algorithm that solves the stable matching problem requires W(?{n/Blogn}){\Omega(\sqrt{n/B\log n})} communication rounds in the worst case, even for graphs of diameter O(log n), where n is the number of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain O(?n){O(\sqrt n)} blocking pairs, and if a pair is considered blocking only if they like each other much more then their assigned match.     

A ρ-calculus of explicit constraint application

Theoretical presentations of the rewriting or ρ-calculus often treat the matching constraint computations as an atomic operation although matching constraints are explicitly expressed. Actual implementations have to take a more realistic view: computations needed in order to find the solutions of a matching equation can have an important impact on the (efficiency of the) calculus for some matching theories and the substitution application usually involves a term traversal. Following the works on explicit substitutions in the λ-calculus, we present two versions of the ρ-calculus, one with explicit matching and one with explicit substitutions, together with a version that combines the two and considers efficiency issues and more precisely the composition of substitutions. The approach is general, allowing for potential extensions to various matching theories. We establish the confluence of the calculus and the termination of the explicit constraint handling and application sub-calculus.     

Complexity of Nilpotent Unification and Matching Problems

We consider the complexity of equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x, x)=0 where 0 is a constant. We show that nilpotent unification and matching are NP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. However, when 0 is also assumed to be the unity for the function f, the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. An application of this result to a subclass of set constraints is illustrated. Second-order matching modulo nilpotence is shown to be undecidable.     

Similarity K-d tree method for sparse point pattern matching with underlying non-rigidity

We propose a method for matching non-affinely related sparse model and data point-sets of identical cardinality, similar spatial distribution and orientation. To establish a one-to-one match, we introduce a new similarity K-dimensional tree. We construct the tree for the model set using spatial sparsity priority order. A corresponding tree for the data set is then constructed, following the sparsity information embedded in the model tree. A matching sequence between the two point sets is generated by traversing the identically structured trees. Experiments on synthetic and real data confirm that this method is applicable to robust spatial matching of sparse point-sets under moderate non-rigid distortion and arbitrary scaling, thus contributing to non-rigid point-pattern matching.     

New primal and dual matching heuristics

We describe a new heuristic for constructing a minimum-cost perfect matching designed for problems on complete graphs whose cost functions satisfy the triangle inequality (e.g., Euclidean problems). The running time for ann node problem is O(n logn) after a minimum-cost spanning tree is constructed. We also describe a procedure which, added to Kruskal's algorithm, produces a lower bound on the size of any perfect matching. This bound is based on a dual problem which has the following geometric interpretation for Euclidean problems: Pack nonoverlapping disks centered at the nodes and moats surrounding odd sets of nodes so as to maximize the sum of the disk radii and moat widths.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Dynamic constraints for record matching

This paper investigates constraints for matching records from unreliable data sources. (a) We introduce a class of matching dependencies (mds) for specifying the semantics of unreliable data. As opposed to static constraints for schema design, mds are developed for record matching, and are defined in terms of similarity predicates and a dynamic semantics. (b) We identify a special case of mds, referred to as relative candidate keys (rcks), to determine what attributes to compare and how to compare them when matching records across possibly different relations. (c) We propose a mechanism for inferring mds, a departure from traditional implication analysis, such that when we cannot match records by comparing attributes that contain errors, we may still find matches by using other, more reliable attributes. Moreover, we develop a sound and complete system for inferring mds. (d) We provide a quadratic-time algorithm for inferring mds and an effective algorithm for deducing a set of high-quality rcks from mds. (e) We experimentally verify that the algorithms help matching tools efficiently identify keys at compile time for matching, blocking or windowing and in addition, that the md-based techniques effectively improve the quality and efficiency of various record matching methods.     

The general maximum matching algorithm of micali and vazirani

We give a clear exposition of the algorithm of Micali and Vazirani for computing a maximum matching in a general graph. This is the most efficient algorithm known for general matching. On a graph withn vertices andm edges this algorithm runs inO(n ^1/2 m) time.Work on this paper has been supported by the Office of Naval Research under Contract N00014-85-K-0570 and by the Eastman Kodak Company.     

Hardness and approximation of minimum maximal matchings

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2?1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2?23/18k) in k-regular graphs admitting a perfect matching.     

Approximate Boyer-Moore String Matching for Small Alphabets

Recently a new variation of approximate Boyer-Moore string matching was presented for the k-mismatch problem. The variation, called FAAST, is specifically tuned for small alphabets. We further improve this algorithm gaining speedups in both preprocessing and searching. We also present three variations of the algorithm for the k-difference problem. We show that the searching time of the algorithms is average-optimal and the preprocessing also has a lower time complexity than FAAST. Our experiments show that our algorithm for the k-mismatch problem is about 30% faster than FAAST and the new algorithms compare well against other state-of-the-art algorithms for approximate string matching.     

An evaluation of medium-grain dataflow code

In this paper, we study several issues related to the medium grain dataflow model of execution. We present bottom-up compilation of medium grainclusters from a fine grain dataflow graph. We compare thebasic block and thedependence sets algorithms that partition dataflow graphs into clusters. For an extensive set of benchmarks we assess the average number of instructions in a cluster and the reduction in matching operations compared with fine grain dataflow execution. We study the performance of medium grain dataflow when several architectural parameters, such as the number of processors, matching cost, and network latency, are varied. The results indicate that medium grain execution offers a good speedup over the fine grain model, that it is scalable, and tolerates network latency and high matching costs well. Medium grain execution can benefit from a higher output bandwidth of a processor and fainally, a simple superscalar processor with an issue rate of two is sufficient to exploit the internal parallelism of a cluster. This work is supported in part by NSF Grants CCR-9010240 and MIP-9113268.     

Scheduling Protocols for Switches with Large Envelopes

Traditionally, switches make scheduling decisions on the granularity of a packet. However, this is becoming increasingly difficult since network bandwidth is growing rapidly whereas packet sizes remain largely unchanged. Therefore the service time of an individual packet is decreasing rapidly. In this paper we study switches that make scheduling decisions on the granularity of an envelope which can be much larger than a packet in size. For an output-queued switch with envelope size E, each output chooses one input every E time steps and transmits packets from this chosen input during the next E steps. For an input-queued switch with envelope size E, one matching from the inputs to the outputs is computed every E steps and only the input–output pairs that are defined by this matching are allowed to transmit packets during the next E steps. Traditional switches correspond to envelope size E = 1 and almost all previous scheduling work deals with this case exclusively. We first show how some stable protocols for scheduling networks of output-queued switches with E = 1 fail for arbitrary E when these protocols are generalized in the most straightforward manner. We then present an extremely simple protocol that does guarantee network stability for output-queued switches for any E ≥ 1. For input-queued switches we first present a max-weight matching protocol that is stable for a single switch with arbitrary E. We then present a more complex protocol that achieves stability for a network of input-queued switches for any E ≥ 1.     

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.Work on this paper has been supported by the Office of Naval Research under Contract N00014-85-K-0570.