Analysis of the Karmarkar-Karp differencing algorithm

Boettcher, Stefan; Mertens, Stephan

doi:10.1140/epjb/e2008-00320-9

Cited by 50 publications

(51 citation statements)

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“…3 Most insights into finite-dimensional systems have been gained through computational approaches that elucidate low-T properties. [4][5][6] Here, we extract the response induced through defect interfaces 7,8 at T = 0, created by fixing the spins along the two faces of the open boundary in one direction. Ground state energies E 0 and E 0 Ј of an instance of size N = L d are determined, which differ by reversing all spins on one of the faces.…”

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confidence: 99%

Low-temperature phase boundary of dilute-lattice spin glasses

Boettcher

Marchetti

2008

Phys. Rev. B

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The thermal-to-percolative crossover exponent , well known for ferromagnetic systems, is studied extensively for Edwards-Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold p c using an improved reduction algorithm. Simulations extend to system sizes above N =10 8 in dimensions d = 2 , . . . , 7. The results can be related to the behavior of the transition temperature T g ϳ͑p − p c ͒ between the paramagnetic and the glassy regime for p p c . In three dimensions, where our simulations predict = 1.127͑5͒, this scaling form for T g provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.The exploration of low-temperature properties of disordered systems remains an important and challenging problem. 1,2 The paradigmatic model for such phenomena is the Edwards-Anderson ͑EA͒ spin glass, 3Disorder effects arise via quenched random bonds, J i,j , mixing ferro-and antiferromagnetic couplings between nearestneighbor spins, which lead to conflicting constraints and frustrated variables. It is believed that an understanding of static and dynamic features of EA may aid a description of the unifying principles expressed in glassy materials. 3 Most insights into finite-dimensional systems have been gained through computational approaches that elucidate low-T properties. [4][5][6] Here, we extract the response induced through defect interfaces 7,8 at T = 0, created by fixing the spins along the two faces of the open boundary in one direction. Ground state energies E 0 and E 0 Ј of an instance of size N = L d are determined, which differ by reversing all spins on one of the faces. The distribution P͑⌬E͒ of interface energies ⌬E = E 0 Ј− E 0 created by this perturbation of scale L on the boundary is obtained. The typical energy scale, represented by the deviation ͑⌬E͒, grows as

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confidence: 99%

Low-temperature phase boundary of dilute-lattice spin glasses

Boettcher

Marchetti

2008

Phys. Rev. B

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“…Unlike the Travelling Salesman Problem, the performance of polynomial-time heuristics, such as the Karmarkar-Karp Differencing Algorithm, is incredibly poor for large instances of the Number Partitioning Problem [17]. In order to try and understand why the NPP is difficult for heuristics, the landscapes of NPP instances have been analysed using techniques from statistical mechanics and traditional problem landscape analysis.…”

Section: Existing Resultsmentioning

confidence: 99%

“…The Number Partitioning Problem (NPP) defined in Appendix A.2.2 is known to undergo phase transitions in terms of the number of global optima, the size of plateaus, and the difficulty of exact solvers [17,22,174]. This section investigates how the length scales of instances of NPP change throughout different stages of the phase transition.…”

Section: Analysis Of Number Partitioning Problemsmentioning

confidence: 99%

“…2.2, is another combinatorial optimization problem that exhibits phase transitions in the difficulty of exact solvers, as well as the number of global optima and size of plateaus [17,22,174]. Instances can be generated at a particular stage of the phase transition through the control parameter, k. …”

Section: Number Partitioning Problemsmentioning

confidence: 99%

“…The NPP is known to undergo a phase transition in a number of landscape properties as well as difficulty for heuristic algorithms [17,22,61,174]. The stage of the phase transition can be controlled using the parameter k = 1 n log 2 m. There is a single critical stage at which the problem structure and difficulty changes drastically.…”

Section: A22 Number Partitioning Problemmentioning

confidence: 99%

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Analysing and comparing problem landscapes for black-box optimization via length scale

Morgan¹

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Optimization problems are of fundamental practical importance and can be found in almost every aspect of human endeavour. Yet remarkably, we have a very limited understanding of the nature of optimization problems and subsequently of how, why and when different algorithms perform well or poorly. The notion of a problem landscape captures the relationship between the objective function and the problem variables. It is clear that the structure of this landscape is vital in understanding optimization problems, however analysis of this structure presents major challenges. Apart from the often high-dimensionality of problem landscapes, information available in the black-box setting is limited to the solutions in the ii

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