Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch‐and‐Cut

Liers, Frauke; Jünger, Michael; Reinelt, Gerhard; Rinaldi, Giovanni

doi:10.1002/3527603794.ch4

Cited by 69 publications

(70 citation statements)

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“…The last two types were d) toroidal 2D and e) toroidal 3D-grid graphs, i.e., 2-and 3-dimensional grid graphs where the first and the last node of each "line" of the grid are adjacent. These instances are relevant, e.g., in Statistical Physics for analyzing the properties of spin glass (see, e.g., [22]). As it is customary in the spin glass literature, for the these two classes of graphs we experimented both with ±1 costs, drawn from a uniform random binary distribution and with costs drawn from a Gaussian distribution.…”

Section: Computational Resultsmentioning

confidence: 99%

New approaches for optimizing over the semimetric polytope

2005

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The semimetric polytope is an important polyhedral structure lying at the heart of hard combinatorial problems. Therefore, linear optimization over the semimetric polytope is crucial for a number of relevant applications. Building on some recent polyhedral and algorithmic results about a related polyhedron, the rooted semimetric polytope, we develop and test several approaches, mainly based over Lagrangian relaxation and application of Non Differentiable Optimization algorithms, for linear optimization over the semimetric polytope. We show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state-of-the-art general purpose linear programming technology. In some cases, good estimates of the dual optimal solution (but not of the primal solution) can be obtained even quicker.

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Section: Computational Resultsmentioning

confidence: 99%

New approaches for optimizing over the semimetric polytope

2005

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“…Each spin was neighbored by the adjacent spin above, below, to the left, and to the right. The coupling constants for neighboring spins were uniformly sampled from {+1, −1}, with all other coupling constants set to 0, thus each instance was drawn from the region of spin glass system space known to contain a comparatively high density of difficult instances [43,44].…”

Section: Ising Spin Glassesmentioning

confidence: 99%

MARLEDA: Effective distribution estimation through Markov random fields

Miikkulainen

Alden

2016

Theoretical Computer Science

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Many problems within the biological sciences, such as DNA sequencing, protein structure prediction, and molecular docking, are being approached computationally. These problems require sophisticated solution methods that understand the complex natures of biological domains. Traditionally, such solution methods are problem specific, but recent advances in generic problem-solvers furnish hope for a new breed of computational tools. The challenge is to develop methods that can automatically learn or acquire an understanding of a complex problem domain.Estimation of Distribution Algorithms (EDAs) are generic search methods that use statistical models to learn the structure of a problem domain.EDAs have been successfully applied to many difficult search problems, such as circuit design, optimizing Ising spin glasses, and various scheduling tasks. Computational biology is a computational science that promises tangible, life-altering benefits. Research in fields such as genomics, protein structure prediction, and molecular docking, finds application in pharmacology and gene therapy. Consequently, day-to-day medicine and health have been improved, with the greatest breakthroughs still to come. These fields directly depend on advances in computer hardware and software, and there is great demand for effective computational problem-solving methods.Research in computational biology produces computational models of real-world objects or phenomena, such as molecular interactions or phylogenetic trees. The computational models can be studied, adjusted, and experimented with to generate predictions about their real-world counterparts. In 2 turn, real-world experiments help form the basis for new computational models. There is an interplay between real-world and computational research; each informs and refines the other. While a computational formulation of real-world problems is useful and convenient, there are drawbacks.The key difficulty of many computational problems, such as DNA sequence alignment, selecting key attributes for data mining, or optimizing antenna design, is that no analytic solution methods exist. There are also problems that do have analytic solutions, but those methods are too computationally expensive to be practical. Without the ability to construct an ideal solution efficiently, we are often reduced to a procedure of guess-and-check.Such a procedure, formally known as search, tries to identity the best solution(s) among a set of candidates by systematically evaluating alternatives.Search algorithms are, in many cases, a relatively easy and effective means of producing near-optimal solutions to difficult problems. However, in complex domains search is often inefficient, taking too long to be practical. This dissertation develops a search method that handles complexity better by statistically modeling the problem domain. ApproachFortunately, even when analytical methods are not available, for many computational problems it is relatively easy to test the "correctness" of a potential solution. For example, t...

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“…A prominent application of max-cut is in determining energy-minimum states, i.e., ground states, of Ising spin glasses. The first exact branch-and-cut approach for its solution was presented in [9] and developed further in [39]. Extending the number of shores to k > 2, maximum k-cuts need to be computed when determining ground states of Potts glasses.…”

Section: Introductionmentioning

confidence: 99%

“…Most versions are known to be NP-hard. Graph partitioning problems have myriad applications in areas as varied as telecommunications network planning [20], VLSI circuit design [9], sports scheduling [45,21], and statistical physics [39].…”

Section: Introductionmentioning

confidence: 99%

“…The web-based Spin Glass Server [4] is especially designed for fast solutions of instances defined on grids that arise in statistical physics [39]. For instance, for a two-dimensional lattice with L ≤ 80 and periodic boundary conditions, one ground-state computation takes less than two minutes on average on a SUN Opteron (2.2 GHz) machine; for 120 2 lattices the computation takes 28 minutes [40].…”

Section: Introductionmentioning

confidence: 99%

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Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch‐and‐Cut

Cited by 69 publications

References 0 publications

New approaches for optimizing over the semimetric polytope

New approaches for optimizing over the semimetric polytope

MARLEDA: Effective distribution estimation through Markov random fields

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

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