Abstract
The structural property of the search graph plays an important role in the success of local search-based metaheuristic algorithms. Magnification is one of the structural properties of the search graph. This study builds the relationship between the magnification of a search graph and the mixing time of Markov Chain (MC) induced by the local search-based metaheuristics on that search space. The result shows that the ergodic reversible Markov chain induced by the local search-based metaheuristics is inversely proportional to magnification. This result indicates that it is desirable to use a search space with large magnification for the optimization problem in hand rather than using any search spaces. The performance of local search-based metaheuristics may be good on such search spaces since the mixing time of the underlying Markov chain is inversely proportional to the magnification of search space. Using these relations, this work shows that MC induced by the Metropolis Algorithm (MA) mixes rapidly if the search graph has a large magnification. This indicates that for any combinatorial optimization problem, the Markov chains associated with the MA mix rapidly i.e., in polynomial time if the underlying search graph has large magnification. The usefulness of the obtained results is illustrated using the 0/1-Knapsack Problem, which is a well-studied combinatorial optimization problem in the literature and is NP-Complete. Using the theoretical results obtained, this work shows that Markov Chains (MCs) associated with the local search-based metaheuristics like random walk and MA for 0/1-Knapsack Problem mixes rapidly.
Highlights
Most of the combinatorial optimization problems in the real world have high computational complexity, which implies there are no known polynomial time algorithms that exist for such optimization problems
In the proposed work we prove that if the search graph has large magnification, the Markov Chain (MC) induced by Metropolis Algorithm (MA) mixes rapidly i.e., in polynomial time
The results show that the mixing time of the MC induced by MA on the search graph is indirectly proportional to magnification of the search graph
Summary
Most of the combinatorial optimization problems in the real world have high computational complexity, which implies there are no known polynomial time algorithms that exist for such optimization problems. Large magnification implies many edges going out from any cut-set in the search graph This property may be profitably when used by metaheuristic algorithms to avoid getting stuck at local optima and to reach global optima. A structural property called magnification [41] of the search graph plays an important role in the rapid mixing of MC induced by local search-based metaheuristics. In the proposed work we prove that if the search graph has large magnification, the MC induced by MA mixes rapidly i.e., in polynomial time. Establishes the relationship between search graph magnification and mixing time of reversible ergodic MC induced by local search-based metaheuristics (Refer Theorem 2); 3. Proved that if the designed search graph has large magnification, for a particular choice of temperature parameter, the MC induced by MA mixes rapidly, i.e., in polynomial time (Refer Corollarys 1 and 2); 4. Applications of the theoretical results obtained are illustrated Using 0/1-Knapsack problem in Section 5 followed by the conclusion
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
