We present a quasi linear-time algorithm for Maximum Matching on distance-hereditary graphs and some of their generalizations. This improves on [Dragan, WG'97], who proposed such an algorithm for the subclass of (tent,hexahedron)-free distance-hereditary graphs. Furthermore, our result is derived from a more general one that is obtained for b-Matching. In the (unit cost) b-Matching problem, we are given a graph G = (V, E) together with a nonnegative integer capacity b v for every vertex v ∈ V. The objective is to assign nonnegative integer weights (x e) e∈E so that: for every v ∈ V the sum of the weights of its incident edges does not exceed b v , and e∈E x e is maximized. We present the first algorithm for solving b-Matching on cographs, distance-hereditary graphs and some of their generalizations in quasi linear time. For that, we use a decomposition algorithm that outputs for any graph G a collection of subgraphs of G with no edge-cutsets inducing a complete bipartite subgraph (a.k.a., splits). The latter collection is sometimes called a split decomposition of G. Furthermore, there exists a generic method in order to design graph algorithms based on split decomposition [Rao, DAM'08]. However, this technique only applies to " localized " problems: for which a " best " partial solution for any given subgraph in a split decomposition can be computed almost independently from the remaining of the graph. Such framework does not apply to matching problems since an augmenting path may cross the subgraphs arbitrarily. We introduce a new technique that somehow captures all the partial solutions for a given union of subgraphs in a split decomposition, in a compact and amenable way for algorithms — assuming some piecewise linear assumption holds on the value of such solutions. The latter assumption is shown to hold for b-Matching. Doing so, we prove that solving b-Matching on any pair G, b can be reduced in quasi linear-time to solving this problem on a collection of smaller graphs: that are obtained from the subgraphs in any split decomposition of G by replacing every vertex with a constant-size module. In particular, if G has a split decomposition where all subgraphs have order at most a fixed k, then we can solve b-Matching for G, b in O((k log 2 k)·(m+n)·log ||b|| 1)-time. This answers an open question of [, SODA'18].
There is a long line of work whose aim is to find more efficiently solvable special cases for Maximum Matching ( e.g. , see @cite_38 @cite_16 @cite_9 @cite_37 @cite_2 @cite_41 @cite_31 @cite_17 @cite_48 @cite_42 @cite_27 @cite_29 @cite_4 ). In particular, there exist linear-time algorithms for computing a maximum matching on trees, cographs @cite_27 and subclasses of distance-hereditary graphs @cite_9 We stress that the work in @cite_9 is focused on a generic method in order to compute maximum matchings. In particular, the complexity of Maximum Matching on general distance-hereditary graphs is not considered. . Furthermore, a natural follow-up of such investigations is to study whether the techniques used in every case can also be applied to larger classes of graphs. This has be done, especially, for the Split & Match technique introduced in @cite_27 for cographs ( e.g. , see @cite_0 @cite_37 @cite_2 where the latter technique has been applied to other graph classes). We also consider this above aspect in our work, as we seek for a Maximum Matching algorithm on distance-hereditary graphs that can be easily generalized to larger classes of graphs that we define in what follows.
@cite_38 @cite_37 @cite_4 @cite_41 @cite_48 @cite_9 @cite_42 @cite_29 @cite_0 @cite_27 @cite_2 @cite_31 @cite_16 @cite_17
1502916507 2097746044 1965299092 2962968573
The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection. Selective families, a weaker variant of superimposed codes [KS64, F92, 197, CR96], have been recently used to design Deterministic Distributed Broadcast (DDB) protocols for unknown radio networks (a radio network is said to be unknown when the nodes know nothing about the network but their own label) [CGGPR00, CGOR00]. We first provide a general almost tight lower bound on the size of selective families. Then, by reverting the selective families - DDB protocols connection, we exploit our lower bound to construct a family of “hard” radio networks (i.e. directed graphs). These networks yield an O(n log D) lower bound on the completion time of DDB protocols that is superlinear (in the size n of the network) even for very small maximum eccentricity D of the network, while all the previous lower bounds (e.g. O(D log n) [CGGPR00]) are superlinear only when D is almost linear. On the other hand, the previous upper bounds are all superlinear in n independently of the eccentricity D and the maximum in-degree d of the network. We introduce a broadcast technique that exploits selective families in a new way. Then, by combining selective families of almost optimal size with our new broadcast technique, we obtain an O(Dd log3 n) upper bound that we prove to be almost optimal when d = O(n D). This exponentially improves over the best known upper bound [CGR00) when D, d = O(polylogn). Furthermore, by comparing our deterministic upper bound with the best known randomized one [BGI87] we obtain a new, rather surprising insight into the real gap between deterministic and randomized protocols. It turns out that this gap is exponential (as discovered in [BGI87]), but only when the network has large maximum in-degree (i.e. d = O(na), for some constant a > O). We then look at the multibroadcast problem on unknown radio networks. A similar connection to that between selective families and (single) broadcast also holds between superimposed codes and multibroadcast. We in fact combine a variant of our (single) broadcast technique with superimposed codes of almost optimal size available in literature [EFF85, HS87, I97, CHI99]. This yields a multibroadcast protocol having completion time O(Dd2 log3 n). Finally, in order to determine the limits of our multibroadcast technique, we generalize (and improve) the best known lower bound [CR96] on the size of superimposed codes. We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial, but far from linear, time. Thus, our goal is to construct algorithms with running time of the form poly(k)·n or poly(k) · n log n, where k is the width of the tree decomposition given on the input. Such procedures would outperform the best known algorithms for the considered problems already for moderate values of the treewidth, like O(n1 c) for some small constant c. Our results include: -- an algorithm for computing the determinant and the rank of an n × n matrix using O(k3 · n) time and arithmetic operations; -- an algorithm for solving a system of linear equations using O(k3 · n) time and arithmetic operations; -- an O(k3 O n log n)-time randomized algorithm for finding the cardinality of a maximum matching in a graph; -- an O(k4 · n log2 n)-time randomized algorithm for constructing a maximum matching in a graph; -- an O(k2 · n log n)-time algorithm for finding a maximum vertex flow in a directed graph. Moreover, we provide an approximation algorithm for treewidth with time complexity suited to the running times as above. Namely, the algorithm, when given a graph G and integer k, runs in time O(k7 · n log n) and either correctly reports that the treewidth of G is larger than k, or constructs a tree decomposition of G of width O(k2). The above results stand in contrast with the recent work of [SODA 2016], which shows that the existence of algorithms with similar running times is unlikely for the problems of finding the diameter and the radius of a graph of low treewidth.