Understand instance-dependent performance of universal solvers

Characterize the instance features and structural properties of submodular and supermodular optimization problems that determine when SuperGreedy++, the Frank–Wolfe algorithm, or the Fujishige–Wolfe minimum-norm-point algorithm performs best in terms of runtime and solution quality across tasks such as Densest Subgraph, Densest Supermodular Set, Unrestricted Sparsest Submodular Set, Unrestricted Densest Supermodular Set, Submodular Function Minimization, and Minimum Norm Point.

Background

The paper demonstrates broad equivalences between several problems (SFM, DSS, USSS, UDSS, MNP) and shows that three algorithms—SuperGreedy++, Frank–Wolfe, and the Fujishige–Wolfe MNP algorithm—can serve as universal solvers across these tasks.

Extensive experiments reveal that while at least one of these algorithms consistently outperforms problem-specific heuristics, the identity of the best-performing method varies by instance. This motivates a theoretical understanding of what structural properties predict which algorithm will excel on a given instance.