Cyclic group and knapsack facets

Juli�n Ar�oz, Lisa Evans, Ralph E. Gomory, Ellis L. Johnson
2003 Mathematical programming  
Any integer program may be relaxed to a group problem. We define the master cyclic group problem and several master knapsack problems, show the relationship between the problems, and give several classes of facet-defining inequalities for each problem, as well as a set of mappings that take facets from one type of master polyhedra to another. For case 2, we partition indices 3, ..., n into six sets: We now construct a set of relations for i in each set that is almost lower-triangular, and give
more » ... simple argument for linear independence. By a set of lower-triangular relations, we mean for a relation i, all coefficients ρ j for j > i are zero in the relation. If i even: 0 = ρ 2 + ρ i−2 − ρ i . J 2 : 0 = 2ρ d+1 − ρ 2(d+1) (this relation violates the lower-triangular property, and we will return to it later). 2 − ρ n To make the relations for J 2 lower triangular, we subtract the relations for i = 2d + 2 and d + 2 to find the new relation 0 = −ρ 2 − 2ρ d + 2ρ d+1 . Although the relations for J 5 are not lower triangular, clearly they are linearly independent from the relation for J 6 because they only have the variable ρ n in common. Corollary 6.4. Given a facet (ρ, ρ n ) of P (K n ) tilted so that ρ n = 0, if ρ 2 = 0, then (ρ, ρ n ) must be a (1, 0, −1) facet, up to multiplication by a constant. Proof. Assume ρ is scaled so that ρ 1 = 1. By subadditivity and ρ 2 = 0, 0 ≥ ρ 2k ≥ ρ 2(k+1) for k = 1, ..., n 2 −1. Also by subadditivity and ρ 1 = 1, 1 ≥ ρ 2k−1 ≥ ρ 2k+1 for
doi:10.1007/s10107-003-0390-x fatcat:jxwdb56vqvfzxetu3a2b72kjhe