A New Contraction Technique with Applications to Congruency-Constrained Cuts

Abstract

Minimum cut problems are among the most classical problems in Combinatorial Optimization and are used in a wide set of applications. Some of the best-known efficiently solvable variants include global minimum cuts, minimum s-t cuts, and minimum odd cuts in undirected graphs. We study a problem class that can be seen to generalize the above variants, namely finding congruency-constrained minimum cuts, i.e., we consider cuts whose number of vertices is congruent to r modulo m, for some integers r and m. Apart from being a natural generalization of odd cuts, congruency-constrained minimum cuts exhibit an interesting link to a long-standing open problem in Integer Programming, namely whether integer programs described by an integer constraint matrix with bounded subdeterminants can be solved efficiently.