Are stable instances easy?

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP–hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polynomial time all sufficiently stable instances of some NP–hard problem. The paper focuses on the Max–Cut problem, for which we show that this is indeed the case.

💡 Research Summary

The paper introduces a novel structural notion called “stability” for discrete optimization problems and investigates whether this property can make otherwise NP‑hard problems tractable on certain instances. For a weighted graph G = (V, E, w), an instance is defined as ε‑stable if, after independently scaling every edge weight by any factor in the interval