Problem
Research questions and friction points this paper is trying to address.
multi-separator polytope
facet characterization
integer linear programming
graph multi-separator problem
polyhedral combinatorics
The canonical facets of multi-separator polytopes
🤖 AI Summary
This study addresses the lack of polyhedral theory for the multi-separator problem in image segmentation by formulating the feasible solution polytope via integer linear programming. Leveraging graph-theoretic conditions, we characterize its facets and strengthen associated inequalities. The work provides a complete description of canonical facets, establishes a totally dual integral formulation for path instances, and reveals geometric projection relationships with Boolean quadric and lifted multicut polytopes. These contributions fill critical theoretical gaps by achieving a comprehensive characterization of multi-separator polytope facets and inequality strengthening. Ultimately, this research offers robust theoretical foundations and novel perspectives for combinatorial optimization in image segmentation, advancing the mathematical understanding of partition-based vision problems through rigorous polyhedral analysis.
📝 Abstract
We initiate a polyhedral study of the graph multi-separator problem proposed by Irmai et al. (2024) as an alternative to the lifted multicut problem for application to the task of image segmentation. Starting with an integer linear program (ILP) formulation and the multi-separator polytope spanned by its feasible solutions, we characterize in terms of efficiently-decidable, graph-theoretic conditions all facets induced by inequalities of the ILP. We proceed by strengthening these inequalities and describing additional facets of some multi-separator polytopes induced by the stronger inequalities. Specifically, we obtain a totally dual integral description of the multi-separator polytope for paths in the case where separation is considered for all vertex pairs. Finally, we relate the multi-separator polytope to the boolean quadric polytope, showing that facets induced by odd-cycle inequalities do not transfer generally, and to the lifted multicut polytope, showing that either polytope is a projection of a face of the other.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Multi-separator polytope
Facet characterization
Totally dual integral
Integer linear programming
Lifted multicut
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