Capturing structure in hard combinatorial problems

Abstract

For many hard combinatorial problems that arise from real-world applications, the conventional theory of algorithms and complexity cannot give reasonable (i.e., polytime) performance guarantees and considers such problems as intractable. Nevertheless, heuristics-based algorithms and solvers work surprisingly well on real-world instances, which suggests that our world may be “friendly enough” to make many typical computational tasks poly-time- challenging the value of the conventional worst-case complexity view in CS (Bart Selman, 2012). Indeed, there is an enormous gap between theoretical performance guarantees and the empirically observed performance of solvers. Efficient solvers exploit the “hidden structure” of real-world problems, and so a theoretical framework that explains practical problem hardness and easiness must not ignore such structural aspects.