Application of statistical mechanics to combinatorial optimization problems: the chromatic number problem and q-partitioning of a graph
Methods of statistical mechanics are applied to two important NP-complete combinatorial optimization problems. The first is the chromatic number problem, which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color. The second is partitioning of a graph into q equal subgraphs so as to minimize intersubgraph connections. Both models are mapped into a frustrated Potts model, which is related to the q-state Potts spin glass. For the first problem, the authors obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation. The quenched model is also discussed. For the second problem they obtain analytic and numerical results by evaluating the groundstate energy of the q = 3 and 4 Potts spin glass using Parisi's replica symmetry breaking. They also perform some numerical simulations to test the theoretical result and obtain very good agreement.
- Research Organization:
- Univ. of Pittsburgh, PA (USA)
- OSTI ID:
- 5372836
- Journal Information:
- J. Stat. Phys.; (United States), Vol. 48:3/4
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
CRYSTAL MODELS
COLOR
COMPUTER GRAPHICS
OPTIMIZATION
SPIN GLASS STATE
STATISTICAL MECHANICS
TOPOLOGICAL MAPPING
ANTIFERROMAGNETISM
COMPUTERIZED SIMULATION
GRAPHS
GROUND STATES
HAMILTONIANS
QUANTUM MECHANICS
SYMMETRY BREAKING
ENERGY LEVELS
MAGNETISM
MAPPING
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MECHANICS
OPTICAL PROPERTIES
ORGANOLEPTIC PROPERTIES
PHYSICAL PROPERTIES
QUANTUM OPERATORS
SIMULATION
TRANSFORMATIONS
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics