235 research outputs found
Exact Ground States of Large Two-Dimensional Planar Ising Spin Glasses
Studying spin-glass physics through analyzing their ground-state properties
has a long history. Although there exist polynomial-time algorithms for the
two-dimensional planar case, where the problem of finding ground states is
transformed to a minimum-weight perfect matching problem, the reachable system
sizes have been limited both by the needed CPU time and by memory requirements.
In this work, we present an algorithm for the calculation of exact ground
states for two-dimensional Ising spin glasses with free boundary conditions in
at least one direction. The algorithmic foundations of the method date back to
the work of Kasteleyn from the 1960s for computing the complete partition
function of the Ising model. Using Kasteleyn cities, we calculate exact ground
states for huge two-dimensional planar Ising spin-glass lattices (up to
3000x3000 spins) within reasonable time. According to our knowledge, these are
the largest sizes currently available. Kasteleyn cities were recently also used
by Thomas and Middleton in the context of extended ground states on the torus.
Moreover, they show that the method can also be used for computing ground
states of planar graphs. Furthermore, we point out that the correctness of
heuristically computed ground states can easily be verified. Finally, we
evaluate the solution quality of heuristic variants of the Bieche et al.
approach.Comment: 11 pages, 5 figures; shortened introduction, extended results; to
appear in Physical Review E 7
Guarding art galleries by guarding witnesses
Let P be a simple polygon. We de ne a witness set W to be a set of points su
h that if any (prospective) guard set G guards W, then it is guaranteed that G guards P . We show that not all polygons admit a nite witness set. If a fi nite minimal witness set exists, then it cannot contain any witness in the interior of P ; all witnesses must lie on the boundary of P , and there
an be at most one witness in the interior of any edge. We give an algorithm to compute a minimal witness set for P in O(n2 log n) time, if such a set exists, or to report the non-existence within the same time bounds. We also outline an algorithm that uses a witness set for P to test whether a (prospective) guard set sees all points in P
On two problems in graph Ramsey theory
We study two classical problems in graph Ramsey theory, that of determining
the Ramsey number of bounded-degree graphs and that of estimating the induced
Ramsey number for a graph with a given number of vertices.
The Ramsey number r(H) of a graph H is the least positive integer N such that
every two-coloring of the edges of the complete graph contains a
monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi
and Trotter states that there exists a constant c(\Delta) such that r(H) \leq
c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The
important open question is to determine the constant c(\Delta). The best
results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are
constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta
\log^2 \Delta}. We improve this upper bound, showing that there is a constant c
for which c(\Delta) \leq 2^{c \Delta \log \Delta}.
The induced Ramsey number r_{ind}(H) of a graph H is the least positive
integer N for which there exists a graph G on N vertices such that every
two-coloring of the edges of G contains an induced monochromatic copy of H.
Erd\H{o}s conjectured the existence of a constant c such that, for any graph H
on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this
conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an
earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in
the exponent.Comment: 18 page
Using the Game of Mastermind to Teach, Practice, and Discuss Scientific Reasoning Skills
The code-breaking game Mastermind, which can be played in minutes at no cost, creates opportunities for students to discuss scientific reasoning, hypothesis-testing, effective experimental design, and sound interpretation of results
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
A survey of -boundedness
If a graph has bounded clique number, and sufficiently large chromatic
number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made
a number of challenging conjectures about this in the early 1980's, which have
remained open until recently; but in the last few years there has been
substantial progress. This is a survey of where we are now
Extended formulations from communication protocols in output-efficient time
Deterministic protocols are well-known tools to obtain extended formulations,
with many applications to polytopes arising in combinatorial optimization.
Although constructive, those tools are not output-efficient, since the time
needed to produce the extended formulation also depends on the number of rows
of the slack matrix (hence, on the exact description in the original space). We
give general sufficient conditions under which those tools can be implemented
as to be output-efficient, showing applications to e.g.~Yannakakis' extended
formulation for the stable set polytope of perfect graphs, for which, to the
best of our knowledge, an efficient construction was previously not known. For
specific classes of polytopes, we give also a direct, efficient construction of
extended formulations arising from protocols. Finally, we deal with extended
formulations coming from unambiguous non-deterministic protocols
On the (parameterized) complexity of recognizing well-covered (r,l)-graphs.
An (r,â)(r,â)-partition of a graph G is a partition of its vertex set into r independent sets and ââ cliques. A graph is (r,â)(r,â) if it admits an (r,â)(r,â)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,â)(r,â)-well-covered if it is both (r,â)(r,â) and well-covered. In this paper we consider two different decision problems. In the (r,â)(r,â)-Well-Covered Graph problem ((r,â)(r,â) wcg for short), we are given a graph G, and the question is whether G is an (r,â)(r,â)-well-covered graph. In the Well-Covered (r,â)(r,â)-Graph problem (wc (r,â)(r,â) g for short), we are given an (r,â)(r,â)-graph G together with an (r,â)(r,â)-partition of V(G) into r independent sets and ââ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for râ„3râ„3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ââ of cliques in an (r,â)(r,â)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,â)(0,â) g problem parameterized by ââ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNPâNP/polycoNPâNP/poly
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