By Ellis L. Johnson
This monograph considers natural integer programming difficulties which difficulty packing, partitioning or masking. For this classification of difficulties, an algorithmic framework utilizing a duality process is available. moreover, the writer proposes for the 1st time a normal framework for either packing and overlaying difficulties characterizing the convex complete of integer ideas.
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Extra info for Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-Group Problems
An essential point is: When does the solution to the group problem (5) satisfy the integer problem? The nonbasic variables XN are given by (5), and the basic variables by Because of (5), XB will be integer, but the nonnegativity requirement in the integer program was dropped so that XB may not satisfy XB ^ 0. Consider the cone 5TB of those y E &tm such that B^y ^ 0. Clearly, b E ft*. Now if b is far enough in the interior of the cone, then will hold for all jc^ satisfying when JN is the index set of nonbasic variables XN.
All components are reduced to their fractional parts. This group is of order Dm while ^ = Zm/Z(B) is only of order D. However, the columns of N will generate a subgroup of order D. The much larger order of ^ may not be of consequence if one is using a method of solving the group problem which only looks at group elements g generated. If two or more columns of N map onto the same group element, then one with least cost can be chosen and the rest set to zero since there is always an optimum group solution with them at value zero.
The automorphism i// maps S onto a matrix i//(5) obtained by adding all of the rows which map onto the same element h G dtf and removing duplicate columns. Now, i//(S) represents the same inequalities 0 ^ Tr'(g') + ir'(h') - ir'(g' + / z ' ) , 0 ^ i r ' ( s ' ) , a n d l ^ 77'^) in %. A facet 77' of &(W, ^(go)) is a solution for which equality holds corresponding to a basis of \jj(S). The TT as defined in the statement of the theorem is easily seen to satisfy the inequalities (l)-(4). It remains to show that the columns of S where equality holds include a basis of S.
Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-Group Problems by Ellis L. Johnson