
An Improved Approximation Algorithm for TSP in the Half Integral Case
We design a 1.49993approximation algorithm for the metric traveling sal...
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Tight complexity lower bounds for integer linear programming with few constraints
We consider the ILP Feasibility problem: given an integer linear program...
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Efficient sequential and parallel algorithms for multistage stochastic integer programming using proximity
We consider the problem of solving integer programs of the form min{ c^βΊ...
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CongruencyConstrained TU Problems Beyond the Bimodular Case
A longstanding open question in Integer Programming is whether integer ...
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Interruptible Algorithms for Multiproblem Solving
In this paper we address the problem of designing an interruptible syste...
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New Bounds for the Vertices of the Integer Hull
The vertices of the integer hull are the integral equivalent to the well...
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Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs
We consider ILPs, where each variable corresponds to an integral point within a polytope π«, i. e., ILPs of the form min{c^β€xβ_pβπ«β©β€^d x_p p = b, xββ€^π«β©β€^d_β₯ 0}. The distance between an optimal fractional solution and an optimal integral solution (called proximity) is an important measure. A classical result by Cook et al.Β (Math. Program., 1986) shows that it is at most Ξ^Ξ(d) where Ξ is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the righthand side b is replaced by another righthand side b'. The distance between an optimal solution x w.r.t.Β b and an optimal solution x' w.r.t.Β b' (called sensitivity) is similarly bounded, i. e., β bb' β_1Β·Ξ^Ξ(d), also shown by Cook et al. Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of Ξ, require negative entries in the constraint matrix, or have fractional righthand sides. Hence, these lower bounds often do not correspond to instances from algorithmic problems. This work presents for each Ξ > 0 and each d > 0 ILPs of the above type with nonnegative constraint matrices such that their proximity and sensitivity is at least Ξ^Ξ(d). Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the configuration ILP. We thereby show that the results of Cook et al. are indeed tight, even for instances arising naturally from problems in combinatorial optimization.
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