Non-standard approaches to integer programming

Vectors in a box

“…Let us suppose, for example, that a < b; then we calculate that the kth coordinate of s − s equals 1 2 tε…”

“…Let s 1 := i∈S 1 v i and s 2 := i∈S 2 v i = s − s 1 . Moreover, let s (1) 1 ∈ Z d and s (2) 1 ∈ Z d be the projections of s 1 onto the coordinates 1 through d and the coordinates d + 1 through 2d, respectively, and similarly for s (1) 2 , s (2) 2 . First let us suppose that (s 1 ) 2d+1 = 0.…”

“…There is a natural and, in our opinion, interesting variant of the quantity τ (d), where one again considers sequences v 1 The IP connection. The quantity τ (d) has been motivated by a connection to an algorithm for integer programming.…”

“…In that remarkable work, which introduced several important ideas of modern polyhedral combinatorics, Gomory defined a certain "universal" polyhedron, the master cyclic group polyhedron, whose faces encode all instances of integer programs in a certain class (see, e.g., [1] for an introduction). Dash et al [5] use the somewhat related concept of the master equality polyhedron K m (N , b), which we recall below.…”

“…and let b ∈ I be a vector corresponding to the right-hand side in (1). Then the master equality polyhedron resides in R I and it is defined as The variables of the linear program are π v for all integer vectors v ∈ I , and the main kind of constraints in it are subadditivity constraints of the form…”