Problem
Research questions and friction points this paper is trying to address.
Forcing set
Anti-forcing set
Global Min Cut
Min s-t Cut
Computational complexity
On the Complexity of Forcing and Anti-Forcing Minimum Cuts
🤖 AI Summary
This study investigates the computational complexity of forced and forbidden sets in minimum cut problems. Focusing on both global and s-t minimum cuts, it systematically examines computational challenges across two variants: those where an optimal cut is provided and those where it is not. Methodologically, the work integrates graph theory with combinatorial optimization, employing algorithm design and NP-completeness reduction techniques. The core contribution lies in precisely delineating the complexity boundaries of these problems by devising efficient algorithms for polynomially solvable cases and rigorously establishing NP-completeness for the remaining ones. Ultimately, this research comprehensively clarifies the computational feasibility of each variant, thereby bridging a significant theoretical gap in the field.
📝 Abstract
For an instance of a combinatorial optimization problem, a \emph{forcing set} is a set of elements such that there is a unique optimal solution including it. Symmetrically, an \emph{anti-forcing set} is a set of elements such that there is a unique optimal solution excluding it. In this paper, we study the problems of computing smallest forcing and anti-forcing sets for two classical cut problems, \textsc{Global Min Cut} and \textsc{Min $s$--$t$ Cut}. We also consider variants in which the optimal cut to be uniquely determined is given as input. For each of these problems, we either give a polynomial-time algorithm or prove \NP-completeness.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Forcing Set
Anti-Forcing Set
Global Min Cut
Min s-t Cut
Computational Complexity
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