On the practically interesting instances of MAXCUT

The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to be desired. In application areas, practically interesting instances very often occupy just a tiny part of an algorithm’s space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following [BL], we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, (1 + )-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time Ω( √ n)-stable instances of MAXCUT, substantially improving the best previously known result. ∗Parasight inc, Agudat sport hapoel 1, Jerusalem, Israel. [email protected] †Department of Mathematics, Hebrew University, Jerusalem 91904, Israel. Supported in part by a binational Israel-USA grant 2008368. [email protected] ‡School of Computer Science and Engineering, Hebrew University, Jerusalem 91904, Israel. Supported in part by a binational Israel-USA grant 2008368. [email protected] §Department of Mathematics, Rutgers University, Piscataway, NJ 08854. Supported in part by NSF under grant CCF-0832787 and by a binational Israel-USA grant 2008368. [email protected]. ar X iv :1 20 5. 48 93 v1 [ cs .C C ] 2 2 M ay 2 01 2