On the Complexity of Forcing and Anti-Forcing Minimum Cuts

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.