Average-case approximation ratio of the 2-opt algorithm for the TSP

We show that the 2-opt heuristic for the traveling salesman problem achieves an expected approximation ratio of roughly for instances with n nodes, where the edge weights are drawn uniformly and independently at random.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.     

Symbolic topological sorting with OBDDs

We present a symbolic OBDD algorithm for topological sorting which requires O(log²|V|) OBDD operations. Then we analyze its true runtime for the directed grid graph and show an upper bound of O(log⁴|V|loglog|V|). This is the first true runtime analysis of a symbolic OBDD algorithm for a fundamental graph problem, and it demonstrates that one can hope that the algorithm behaves well for sufficiently structured inputs.     

Maintaining dynamic minimum spanning trees: An experimental study

We report our findings on an extensive empirical study on the performance of several algorithms for maintaining minimum spanning trees in dynamic graphs. In particular, we have implemented and tested several variants of the polylogarithmic algorithm by Holm et al., sparsification on top of Frederickson’s algorithm, and other (less sophisticated) dynamic algorithms. In our experiments, we considered as test sets several random, semi-random and worst-case inputs previously considered in the literature together with inputs arising from real-world applications (e.g., a graph of the Internet Autonomous Systems).     

Colouring random graphs

We discuss some results concerned with the behaviour of colouring algorithms on large random graphs.     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

Random triangle removal

Starting from a complete graph on n   vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollobás and Erd?s (1990) conjectured that the expected final number of edges has order n^3/2$n^{3/2}$. An upper bound of o(n²)$o(n^2)$ was shown by Spencer (1995) and independently by Rödl and Thoma (1996). Several bounds were given for variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald (1999)), while the best known upper bound for the original question of Bollobás and Erd?s was n^7/4+o(1)$n^{7/4+o(1)}$ due to Grable (1997). No nontrivial lower bound was available.     

The degree of a typical vertex in generalized random intersection graph models

In this paper we consider the degree of a typical vertex in two models of random intersection graphs introduced in [E. Godehardt, J. Jaworski, Two models of random intersection graphs for classification, in: M. Schwaiger, O. Opitz (Eds.), Exploratory Data Analysis in Empirical Research, Proceedings of the 25th Annual Conference of the Gesellschaft für Klassifikation e.V., University of Munich, March 14-16, 2001, Springer, Berlin, Heidelberg, New York, 2002, pp. 67-81], the active and passive models. The active models are those for which vertices are assigned a random subset of a list of objects and two vertices are made adjacent when their subsets intersect. We prove sufficient conditions for vertex degree to be asymptotically Poisson as well as closely related necessary conditions. We also consider the passive model of intersection graphs, in which objects are vertices and two objects are made adjacent if there is at least one vertex in the corresponding active model “containing” both objects. We prove a necessary condition for vertex degree to be asymptotically Poisson for passive intersection graphs.     

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献   

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.     

Dynamically maintaining split graphs

We present an algorithm that supports operations for modifying a split graph by adding edges or vertices and deleting edges, such that after each modification the graph is repaired to become a split graph in a minimal way. In particular, if the graph is not split after the modification, the algorithm computes a minimal, or if desired even a minimum, split completion or deletion of the modified graph. The motivation for such operations is similar to the motivation for fully dynamic algorithms for particular graph classes. In our case we allow all modifications to the graph and repair, rather than allowing only the modifications that keep the graph split. Fully dynamic algorithms of the latter kind are known for split graphs [L. Ibarra, Fully dynamic algorithms for chordal graphs and split graphs, Technical Report DCS-262-IR, University of Victoria, Canada, 2000].Our results can be used to design linear time algorithms for some recognition and completion problems, where the input is supplied in an on-line fashion.     

Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph (k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems.For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.     

Two methods for the generation of chordal graphs

In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on new results concerning the dynamic maintenance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. Theoretical and experimental results are presented. In both methods, chordality is preserved along the whole generation process. L. Markenzon’s research is partially supported by grant 301068/2003-8, CNPq, Brazil.     

A probabilistic analysis of some greedy cardinality matching algorithms

This paper deals with the expected cardinality of greedy matchings in random graphs. Different versions of the greedy heuristic for the cardinality matching problem are considered. Experimental data and some theoretical results are reported.