Markov Approximation for Combinatorial Network Optimization

• Published in: IEEE transactions on information theory Vol. 59; no. 10; pp. 6301 - 6327
• Main Authors: Chen, M, Liew, SC, Shao, Z, Kai, C
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Approximation; Approximation algorithms; Approximation methods; Combinatorial optimization; Distributed algorithms; Entropy; Exact sciences and technology; Information theory; Information, signal and communications theory; Markov analysis; Markov approximation; Markov processes; Multiaccess communication; Optimization; Optimization algorithms; Probability distribution; Telecommunications and information theory; time-reversible Markov chains; wireless networks; Performance evaluation; time-reversible Markov chains; Probabilistic approach; Combinatorial problem; Wireless telecommunication; Markov approximation; wireless networks; Probability distribution; Combinatorial optimization; Markov chain; distributed algorithms; Distributed algorithm; Wireless network
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2268923

Many important network design problems are fundamentally combinatorial optimization problems. A large number of such problems, however, cannot readily be tackled by distributed algorithms. The Markov approximation framework studied in this paper is a general technique for synthesizing distributed algorithms. We show that when using the log-sum-exp function to approximate the optimal value of any combinatorial problem, we end up with a solution that can be interpreted as the stationary probability distribution of a class of time-reversible Markov chains. Selected Markov chains among this class yield distributed algorithms that solve the log-sum-exp approximated combinatorial network optimization problem. By examining three applications, we illustrate that the Markov approximation technique not only provides fresh perspectives to existing distributed solutions, but also provides clues leading to the construction of new distributed algorithms in various domains with provable performance. We believe the Markov approximation techniques will find applications in many other network optimization problems.