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Integer programming (IP) is an important and challenging problem. Approximate methods have shown promising performance on both effectiveness and efficiency for solving the IP problem. However, we observed that a large fraction of variables solved by some iterative approximate methods fluctuate around their final converged discrete states in very long iterations. Inspired by this observation, we aim to accelerate these approximate methods by early fixing these fluctuated variables to their converged states while not significantly harming the solution accuracy. To this end, we propose an early fixing framework along with the approximate method. We formulate the whole early fixing process as a Markov decision process, and train it using imitation learning. A policy network will evaluate the posterior probability of each free variable concerning its discrete candidate states in each block of iterations. Specifically, we adopt the powerful multi-headed attention mechanism in the policy network. Extensive experiments on our proposed early fixing framework are conducted to three different IP applications: constrained linear programming, MRF energy minimization and sparse adversarial attack. The former one is linear IP problem, while the latter two are quadratic IP problems. We extend the problem scale from regular size to significantly large size. The extensive experiments reveal the competitiveness of our early fixing framework: the runtime speeds up significantly, while the solution quality does not degrade much, even in some cases it is available to obtain better solutions. Our proposed early fixing framework can be regarded as an acceleration extension of ADMM methods for solving integer programming. The source codes are available at \url{https://github.com/SCLBD/Accelerated-Lpbox-ADMM}.