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

Abstract

We propose a two-phase systematical framework for approximation algorithm design and analysis via Lyapunov function. The first phase consists of using Lyapunov function as an input and outputs a continuous-time approximation algorithm with a provable approximation ratio. The second phase then converts this continuous-time algorithm to a discrete-time algorithm with almost the same approximation ratio along with provable time complexity. One distinctive feature of our framework is that we only need to know the parametric form of the Lyapunov function whose complete specification will not be decided until the end of the first phase by maximizing the approximation ratio of the continuous-time algorithm. Some immediate benefits of the Lyapunov function approach include: (i) unifying many existing algorithms; (ii) providing a guideline to design and analyze new algorithms; and (iii) offer new perspectives to potentially improve existing algorithms. We use various submodular maximization problems as running examples to illustrate our framework.

Online Weakly DR-Submodular Optimization Under Stochastic Cumulative Constraints

“…where is the approximation ratio, and reprensents the loss term [4][5][6][7][8][9][10][11][12] . It is natural to introduce certain factor for the comparator in the online setting, that is, to measure the following:…”