65K05 Mathematical programming methods [See also 90Cxx]
Refine
Document Type
- ZIB-Report (4)
Language
- English (4)
Has Fulltext
- yes (4)
Is part of the Bibliography
- no (4)
Keywords
- Integer Programming (2)
- Network Performance (2)
- Operation Research (2)
- Resource Allocation (2)
- Routing (2)
- Scheduling (2)
- Wireless ad hoc networks (2)
- 1/2}-Chvatal-Gomory cuts (1)
- Kantorovich Theorem (1)
- Linear complementarity problem (1)
Institute
- ZIB Allgemein (4) (remove)
Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
Relaying is a protocol extension for cellular wireless computer networks; in order to utilize radio resources more efficiently, several hops are allowed within one cell. This paper investigates the principle potential of relaying by casting transmission scheduling as a mathematical optimization problem, namely, a linear program. We analyze the throughput gains showing that, irrespective of the concrete scheduling algorithm, performance gains of up to 30\% on average for concrete example networks are achievable.
Relaying -- allowing multiple wireless hops -- is a protocol extension for cellular networks conceived to improve data throughput. Its benefits have only been quantified for small example networks. For assessing its general potential, we define a complex resource allocation\slash{}scheduling problem. Several mathematical models are presented for this problem; while a time-expanded MIP approach turns out intractable, a sophisticated column generation scheme leads to good computational results. We thereby show that for selected cases relaying can increase data throughput by 30\% on the average.
A path following algorithm for linear complementarity problems is presented. Given a point $z$ that approximates a point $z(\tau)$ on the central path with complementarity gap $\tau$, one determines a parameter $\theta\in (0,1)$ so that this point satisfies the hypothesis of the affine invariant Kantorovich Theorem for the equation defining $z((1-\theta)\tau)$. It is shown that $\theta$ is bounded below by a multiple of $n^{-1/2}$, where $n$ is the dimension of the problem. Since the hypothesis of of the Kantorovich Theorem is satisfied the sequence generated by Newton's method, or by the simplified Newton method, will converge to $z((1-\theta)\tau)$. We show that the number of steps required to obtain an acceptable approximation of $z((1-\theta)\tau)$ is bounded above by a number independent of $n$. Therefore the algorithm has $O(\sqrt{n}L)$-iteration complexity. The parameters of the algorithm can be determined in such a way that only one Newton step is needed each time the complementarity gap is decreased.