$\ell_p$-Box ADMM: A Versatile Framework for Integer Programming

This paper revisits the integer programming (IP) problem, which plays a fundamental role in many computer vision and machine learning applications. The literature abounds with many seminal works that address this problem, some focusing on continuous approaches (e.g. linear program relaxation) while others on discrete ones (e.g., min-cut). However, a limited number of them are designed to handle the general IP form and even these methods cannot adequately satisfy the simultaneous requirements of accuracy, feasibility, and scalability. To this end, we propose a novel and versatile framework called $\ell_p$-box ADMM, which is based on two parts. (1) The discrete constraint is equivalently replaced by the intersection of a box and a $(n-1)$-dimensional sphere (defined through the $\ell_p$ norm). (2) We infuse this equivalence into the ADMM (Alternating Direction Method of Multipliers) framework to handle these continuous constraints separately and to harness its attractive properties. More importantly, the ADMM update steps can lead to manageable sub-problems in the continuous domain. To demonstrate its efficacy, we consider an instance of the framework, namely $\ell_2$-box ADMM applied to binary quadratic programming (BQP). Here, the ADMM steps are simple, computationally efficient, and theoretically guaranteed to converge to a KKT point. We demonstrate the applicability of $\ell_2$-box ADMM on three important applications: MRF energy minimization, graph matching, and clustering. Results clearly show that it significantly outperforms existing generic IP solvers both in runtime and objective. It also achieves very competitive performance vs. state-of-the-art methods specific to these applications.