Complexity and approximations for submodular minimization problems on two variables per inequality constraints

Abstract

We demonstrate here that submodular minimization (SM) problems subject to constraints containing up to two variables per inequality, SM2, are 2-approximable in polynomial time and a better approximation factor cannot be achieved in polynomial time unless NP=P. The 2-approximation holds when either the constraints have the round-up property (the rounding up of a feasible fractional solution is feasible) or, if the constraints do not have this property, for monotone submodular functions. The submodular minimization or supermodular maximization on constraints where the coefficients of the two variables in each constraint are of opposite signs (monotone constraints) is solvable in polynomial time. The 2-approximability and the polynomial time solvability for monotone constraints hold also for multi-sets that contain elements with integer multiplicity greater than 1, except that the running time is then pseudo-polynomial in that it depends on the range of the variables. This complexity cannot be improved unless NP=P.Our results indicate that SM2 problems are not much harder than the respective linear integer problems on two variables per constraint: For monotone constraints both problems are polynomial time solvable, and for non-monotone NP-hard problems, both problems have 2-approximation algorithms. For SM2 problems the factor 2 approximation is provably best possible, whereas for the respective linear integer problems it has not been established that the factor 2 is best possible, but this has been conjectured. On the other hand, for SM2 problems where the two variables constraints’ coefficients form a totally unimodular constraint matrix, the linear integer optimization problem is solved in polynomial time, whereas the submodular optimization is proved here to be NP-hard.The submodular minimization NP-hard problems for which our general purpose 2-approximation algorithm applies include submodular-vertex cover, submodular-2SAT, submodular-min satisfiability, submodular-edge deletion for clique, submodular-node deletion for biclique and others.