534 research outputs found
Ramsey numbers R(K3,G) for graphs of order 10
In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12
005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for
which we could not determine the Ramsey number. Most likely these graphs need
approaches focusing on each individual graph in order to determine their
triangle Ramsey number. The results were obtained by combining new
computational and theoretical results. We also describe an optimized algorithm
for the generation of all maximal triangle-free graphs and triangle Ramsey
graphs. All Ramsey numbers up to 30 were computed by our implementation of this
algorithm. We also prove some theoretical results that are applied to determine
several triangle Ramsey numbers larger than 30. As not only the number of
graphs is increasing very fast, but also the difficulty to determine Ramsey
numbers, we consider it very likely that the table of all triangle Ramsey
numbers for graphs of order 10 is the last complete table that can possibly be
determined for a very long time.Comment: 24 pages, submitted for publication; added some comment
An algorithmic approach for multi-color Ramsey graphs
The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45
Ramsey numbers R(K3, G) for graphs of order 10
In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time
An adaptive prefix-assignment technique for symmetry reduction
This paper presents a technique for symmetry reduction that adaptively
assigns a prefix of variables in a system of constraints so that the generated
prefix-assignments are pairwise nonisomorphic under the action of the symmetry
group of the system. The technique is based on McKay's canonical extension
framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the
technique are (i) adaptability---the prefix sequence can be user-prescribed and
truncated for compatibility with the group of symmetries; (ii)
parallelizability---prefix-assignments can be processed in parallel
independently of each other; (iii) versatility---the method is applicable
whenever the group of symmetries can be concisely represented as the
automorphism group of a vertex-colored graph; and (iv) implementability---the
method can be implemented relying on a canonical labeling map for
vertex-colored graphs as the only nontrivial subroutine. To demonstrate the
practical applicability of our technique, we have prepared an experimental
open-source implementation of the technique and carry out a set of experiments
that demonstrate ability to reduce symmetry on hard instances. Furthermore, we
demonstrate that the implementation effectively parallelizes to compute
clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie
Solving Hard Graph Problems with Combinatorial Computing and Optimization
Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are -hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest.
Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number , defined as the smallest order of a -free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were . We improve the upper bound to using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number , which is the smallest such that any -vertex graph contains a cycle of length four or an independent set of order . With the help of combinatorial algorithms, we determine and using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic
On (3, k) Ramsey graphs: Theoretical and computational results
A (3,k,n,e) Ramsey graph is a triangle-free graph on n vertices with e edges and no independent set of size k. Similarly, a (3,k)-, (3,k,n)- or (3,k,n,e)-graph is a (3,k,n,e) Ramsey graph for some nand e. In the first part of the paper we derive an explicit formula for the minimum number of edges in any (3,k,n)graph for n ≤ 3(k-I), i.e. a partial formula for the function e(3,k,n) investigated in [3,5,7]. We prove some general properties of minimum (3,k,n)- graphs with e(3,k,n) edges and present a construction of minimum (3,k+I,3k-I,5k-5)-graphs for k ≥ 2 and minimum (3,k+1,3k,5k)-graphs for k ≥ 4. In the second part of the paper we describe a catalogue of small Ramsey graphs: all (3,k)-graphs for k ~6 and some (3,7)-graphs including all 191 (3,7,22)-graphs, produced by a computer. We present for k ≤ 7 all minimum (3,k,n)-graphs and all 10 maximum (3,7,22)-graphs with 66 edges. *Please refer to full-text for correct equations and numerical value
Smooth Approximations and Relational Width Collapses
We prove that relational structures admitting specific polymorphisms (namely,
canonical pseudo-WNU operations of all arities ) have low relational
width. This implies a collapse of the bounded width hierarchy for numerous
classes of infinite-domain CSPs studied in the literature. Moreover, we obtain
a characterization of bounded width for first-order reducts of unary structures
and a characterization of MMSNP sentences that are equivalent to a Datalog
program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In
particular, the bounded width hierarchy collapses in those cases as well
Smooth Approximations and Relational Width Collapses
We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ? 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well
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