205,239 research outputs found
Construction of near-optimal vertex clique covering for real-world networks
We propose a method based on combining a constructive and a bounding heuristic to solve the vertex clique covering problem (CCP), where the aim is to partition the vertices of a graph into the smallest number of classes, which induce cliques. Searching for the solution to CCP is highly motivated by analysis of social and other real-world networks, applications in graph mining, as well as by the fact that CCP is one of the classical NP-hard problems. Combining the construction and the bounding heuristic helped us not only to find high-quality clique coverings but also to determine that in the domain of real-world networks, many of the obtained solutions are optimal, while the rest of them are near-optimal. In addition, the method has a polynomial time complexity and shows much promise for its practical use. Experimental results are presented for a fairly representative benchmark of real-world data. Our test graphs include extracts of web-based social networks, including some very large ones, several well-known graphs from network science, as well as coappearance networks of literary works' characters from the DIMACS graph coloring benchmark. We also present results for synthetic pseudorandom graphs structured according to the Erdös-Renyi model and Leighton's model
GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs
We present a prototype of a software tool for exploration of multiple
combinatorial optimisation problems in large real-world and synthetic complex
networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial
Explorer), provides a unified framework for scalable computation and
presentation of high-quality suboptimal solutions and bounds for a number of
widely studied combinatorial optimisation problems. Efficient representation
and applicability to large-scale graphs and complex networks are particularly
considered in its design. The problems currently supported include maximum
clique, graph colouring, maximum independent set, minimum vertex clique
covering, minimum dominating set, as well as the longest simple cycle problem.
Suboptimal solutions and intervals for optimal objective values are estimated
using scalable heuristics. The tool is designed with extensibility in mind,
with the view of further problems and both new fast and high-performance
heuristics to be added in the future. GraphCombEx has already been successfully
used as a support tool in a number of recent research studies using
combinatorial optimisation to analyse complex networks, indicating its promise
as a research software tool
Exact Algorithms for Maximum Independent Set
We show that the maximum independent set problem (MIS) on an -vertex graph
can be solved in time and polynomial space, which even is
faster than Robson's -time exponential-space algorithm
published in 1986. We also obtain improved algorithms for MIS in graphs with
maximum degree 6 and 7, which run in time of and
, respectively. Our algorithms are obtained by using fast
algorithms for MIS in low-degree graphs in a hierarchical way and making a
careful analyses on the structure of bounded-degree graphs
New Computational Upper Bounds for Ramsey Numbers R(3,k)
Using computational techniques we derive six new upper bounds on the
classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <=
68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are
improvements by one over the previously best known bounds.
Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on
n vertices without independent sets of order k. The new upper bounds on R(3,k)
are obtained by completing the computation of the exact values of e(3,k,n) for
all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower
bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The
enumeration of all graphs witnessing the values of e(3,k,n) is completed for
all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35
vertices is unique up to isomorphism. For the case of R(3,10), first we
establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently,
that if R(3,10) = 43 then every critical graph is regular of degree 9. Then,
using computations, we disprove the existence of the latter, and thus show that
R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for
R(3,11); added some note
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
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