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We propose a two-phase systematical framework for approximation algorithmdesign and analysis via Lyapunov function. The first phase consists of usingLyapunov function as an input and outputs a continuous-time approximationalgorithm with a provable approximation ratio. The second phase then convertsthis continuous-time algorithm to a discrete-time algorithm with almost thesame approximation ratio along with provable time complexity. One distinctivefeature of our framework is that we only need to know the parametric form ofthe Lyapunov function whose complete specification will not be decided untilthe end of the first phase by maximizing the approximation ratio of thecontinuous-time algorithm. Some immediate benefits of the Lyapunov functionapproach include: (i) unifying many existing algorithms; (ii) providing aguideline to design and analyze new algorithms; and (iii) offering newperspectives to potentially improve existing algorithms. We use varioussubmodular maximization problems as running examples to illustrate ourframework.

Lyapunov function approach for approximation algorithm design and analysis: with applications in submodular maximization