Approximating integer programs with positive right-hand sides

1 Department of Computer and Information S ien e, Linköpings universitet, SE-581 83 Linköping, Sweden 2 É ole polyte hnique, Laboratoire d'informatique (LIX), 91128 Palaiseau Cedex, Fran e Abstra t. We study minimisation of integer linear programs with positive right-hand sides. We show that su h programs an be approximated within the maximum absolute row sum of the onstraint matrix A whenever the variables are allowed to take values in N. This result is optimal under the unique games onje ture. When the variables are restri ted to bounded domains, we show that nding a feasible solution is NP-hard in almost all ases. 1 Introdu tion We study the approximability of minimising integer linear programs with positive right-hand sides. Let n and m be positive integers, representing the number of variables and the number of inequalities, respe tively. Let x = (x1, . . . , xn) be a ve tor of n variables, A be an integerm×n matrix, b ∈ (Z+)m, and c ∈ (Q+∪{0})n. Finally, let X be some given subset of Nn. We onsider here various restri tions of the following integer linear program: Minimise cx subje t to Ax ≥ b, x ∈ X. (IP) Typi ally, X is either N or {x ∈ Z | 0 ≤ x ≤ d} for some d ∈ (Z+)n, where the inequalities are to hold omponentwise. A ommonly o urring instan e of the latter ase is when X = {0, 1}n, soalled 0-1 programming. In all but very restri ted ases, (IP) is