21 research outputs found
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
On edge disjoint spanning trees in a randomly weighted complete graph
Assume that the edges of the complete graph are given independent
uniform edges weights. We consider the expected minimum total weight
of edge disjoint spanning trees. When is large we show
that . Most of the paper is concerned with the case . We
show that \m_2 tends to an explicitly defined constant and that .Comment: Fixed minor issue
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Dynamic Algorithms for Shortest Paths and Matching
There is a long history of research in theoretical computer science devoted to designing efficient algorithms for graph problems. In many modern applications the graph in question is changing over time, and we would like to avoid rerunning our algorithm on the entire graph every time a small change occurs. The evolving nature of graphs motivates the dynamic graph model, in which the goal is to minimize the amount of work needed to reoptimize the solution when the graph changes. There is a large body of literature on dynamic algorithms for basic problems that arise in graphs. This thesis presents several improved dynamic algorithms for two fundamental graph problems: shortest paths, and matching
Survey of local algorithms
A local algorithm is a distributed algorithm that runs in constant time, independently of the size of the network. Being highly scalable and fault-tolerant, such algorithms are ideal in the operation of large-scale distributed systems. Furthermore, even though the model of local algorithms is very limited, in recent years we have seen many positive results for non-trivial problems. This work surveys the state-of-the-art in the field, covering impossibility results, deterministic local algorithms, randomised local algorithms, and local algorithms for geometric graphs.Peer reviewe
GraphMineSuite: Enabling High-Performance and Programmable Graph Mining Algorithms with Set Algebra
We propose GraphMineSuite (GMS): the first benchmarking suite for graph
mining that facilitates evaluating and constructing high-performance graph
mining algorithms. First, GMS comes with a benchmark specification based on
extensive literature review, prescribing representative problems, algorithms,
and datasets. Second, GMS offers a carefully designed software platform for
seamless testing of different fine-grained elements of graph mining algorithms,
such as graph representations or algorithm subroutines. The platform includes
parallel implementations of more than 40 considered baselines, and it
facilitates developing complex and fast mining algorithms. High modularity is
possible by harnessing set algebra operations such as set intersection and
difference, which enables breaking complex graph mining algorithms into simple
building blocks that can be separately experimented with. GMS is supported with
a broad concurrency analysis for portability in performance insights, and a
novel performance metric to assess the throughput of graph mining algorithms,
enabling more insightful evaluation. As use cases, we harness GMS to rapidly
redesign and accelerate state-of-the-art baselines of core graph mining
problems: degeneracy reordering (by up to >2x), maximal clique listing (by up
to >9x), k-clique listing (by 1.1x), and subgraph isomorphism (by up to 2.5x),
also obtaining better theoretical performance bounds
Decompositions of graphs and hypergraphs
This thesis contains various new results in the areas of design theory and edge decompositions of graphs and hypergraphs. Most notably, we give a new proof of the existence conjecture, dating back to the 19th century.
For -graphs and , an -decomposition of G is a collection of edge-disjoint copies of F in G covering all edges of . In a recent breakthrough, Keevash proved that every sufficiently large quasirandom -graph G has a -decomposition (subject to necessary divisibility conditions), thus proving the existence conjecture.
We strengthen Keevash's result in two major directions: Firstly, our main result applies to decompositions into any -graph , which generalises a fundamental theorem of Wilson to hypergraphs. Secondly, our proof framework applies beyond quasirandomness, enabling us e.g. to deduce a minimum degree version.
For graphs, we investigate the minimum degree setting further. In particular, we determine the decomposition threshold' of every bipartite graph, and show that the threshold of cliques is equal to its fractional analogue.
We also present theorems concerning optimal path and cycle decompositions of quasirandom graphs.
This thesis is based on joint work with Daniela Kuhn and Deryk Osthus, Allan Lo and Richard Montgomery