The Support of Integer Optimal Solutions

Abstract

The support of a vector is the number of nonzero-components. We show that given an integral m × n matrix A, the integer linear optimization problem max c T x : Ax = b, x ≥ 0, x ∈ Z n has an optimal solution whose support is bounded by 2m log(2 √ m A ∞ ), where A ∞ is the largest absolute value of an entry of A. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.

Optimizing Sparsity over Lattices and Semigroups

“…There is still little understanding of discrete signals in the compressed sensing paradigm, despite the fact that there are many applications in which the signal is known to have discrete-valued entries, for instance, in wireless communication [22] and the theory of error-correcting codes [7]. Sparsity was also investigated in integer optimization [1,10,20], where many combinatorial optimization problems have useful interpretations as sparse semigroup problems. For example, the edge-coloring problem can be seen as a problem in the semigroup generated by matchings of the graph [18].…”

Sparsity of Integer Solutions in the Average Case

“…q d , −δ . The matrixà has d + 1 columns, so σ asy (Ã) ≤ 1 + d. The matrixà is similar to the example in [1,Theorem 2] and the theory of so-called primorials. We claim if b ∈ Z <0 and b ≡ 1 mod δ, then P (Ã, b) = ∅ and σ(Ã, b) = 1 + d.…”

“…It turns out that the previous upper bound is close to the true value of σ(A). In fact, for every ǫ > 0, Aliev et al [1] provide an example of A for which m log 2 ( A ∞ ) 1/(1+ǫ) ≤ σ(A).…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…Similar results have been used, for instance, for solving of bin-packing problems, see e.g., [154,188]. See [13] for an application to the sparsity of optimal solutions and tighter bounds for special cases such as knapsack problems.…”