2,610 research outputs found
UMSL Bulletin 2023-2024
The 2023-2024 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1088/thumbnail.jp
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On the Power of Threshold-Based Algorithms for Detecting Cycles in the CONGEST Model
It is known that, for every , -freeness can be decided by a
generic Monte-Carlo algorithm running in rounds in the
CONGEST model. For , faster Monte-Carlo algorithms do exist,
running in rounds, based on upper bounding the number of
messages to be forwarded, and aborting search sub-routines for which this
number exceeds certain thresholds. We investigate the possible extension of
these threshold-based algorithms, for the detection of larger cycles. We first
show that, for every , there exists an infinite family of graphs
containing a -cycle for which any threshold-based algorithm fails to detect
that cycle. Hence, in particular, neither -freeness nor
-freeness can be decided by threshold-based algorithms. Nevertheless,
we show that -freeness can still be decided by a
threshold-based algorithm, running in rounds,
which is faster than using the generic algorithm, which would run in
rounds. Moreover, we exhibit an
infinite collection of families of cycles such that threshold-based algorithms
can decide -freeness for every in this collection.Comment: to be published in SIROCCO 202
UMSL Bulletin 2022-2023
The 2022-2023 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1087/thumbnail.jp
Thick Forests
We consider classes of graphs, which we call thick graphs, that have their
vertices replaced by cliques and their edges replaced by bipartite graphs. In
particular, we consider the case of thick forests, which are a subclass of
perfect graphs. We show that this class can be recognised in polynomial time,
and examine the complexity of counting independent sets and colourings for
graphs in the class. We consider some extensions of our results to thick graphs
beyond thick forests.Comment: 40 pages, 19 figure
Taylor University Catalog 2023-2024
The 2023-2024 academic catalog of Taylor University in Upland, Indiana.https://pillars.taylor.edu/catalogs/1128/thumbnail.jp
A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.
Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph
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