Improving the Cook et al. Proximity Bound Given Integral Valued Constraints

  title={Improving the Cook et al. Proximity Bound Given Integral Valued Constraints},
  author={Marcel Celaya and Stefan Kuhlmann and Joseph Paat and Robert Weismantel},
Consider a linear program of the form max cx : Ax ≤ b, where A is an m × n integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardos proved that, given an optimal solution x, if an optimal integral solution z exists, then it may be chosen such that ‖x − z‖ ∞ < n∆, where ∆ is the largest magnitude of any subdeterminant of A. Since then an open question has been to improve this bound, assuming that b is integral valued too. In this manuscript we show that n∆ can be replaced with n/2… 
1 Citations
A Colorful Steinitz Lemma with Applications to Block Integer Programs
The Steinitz constant in dimension d is the smallest value c(d) such that for any norm on R and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of


Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma
The Steinitz lemma is used to show that the ℓ1-distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by m ⋅ (2,m⋅ Δ +1)m.
Proximity in concave integer quadratic programming
The key observation is that, in this setting, proximity phenomena still occur, but only if the authors consider also approximate solutions instead of optimal solutions only, and no upper bound can be given as a function of n and Δ .
Improving Proximity Bounds Using Sparsity
This work bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints.
Distances of optimal solutions of mixed-integer programs
A classic result of Cook et al. (1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is
Sensitivity theorems in integer linear programming
We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to
Polynomial upper bounds on the number of differing columns of $\Delta$-modular integer programs
We consider integer programs (IP) defined by equations and box constraints, where it is known that determinants in the constraint matrix are one measure of complexity. For example, Artmann et al.
Congruency-Constrained TU Problems Beyond the Bimodular Case
In particular, it is shown how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m = 3, and techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation.
Distances to lattice points in knapsack polyhedra
An optimal upper bound for the distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point is given and it is shown that the upper bound can be significantly improved on average.
Convex separable optimization is not much harder than linear optimization
A general-purpose algorithm for converting procedures that solves linear programming problems that is polynomial for constraint matrices with polynomially bounded subdeterminants and an algorithm for finding a ε-accurate optimal continuous solution to the nonlinear problem.
Some proximity and sensitivity results in quadratic integer programming
It is proved, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb′, respectively, depends linearly on ‖b−b′‖1.