On the analysis of optimization problems in arc-dependent networks

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
P. Wojciechowski , M. Williamson , K. Subramani
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引用次数: 0

Abstract

This paper is concerned with the design and analysis of algorithms for optimization problems in arc-dependent networks. A network is said to be arc-dependent if the cost of an arc a depends upon the arc taken to enter a. These networks are fundamentally different from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. We first study the arc-dependent shortest path (ADSP) problem, which is also known as the suffix-1 path-dependent shortest path problem in the literature. This problem has a polynomial time solution if the shortest paths are not required to be simple. The ADSP problem finds applications in a number of domains, including highway engineering, turn penalties and prohibitions, and fare rebates. In this paper, we are interested in the ADSP problem when restricted to simple paths. We call this restricted version the simple arc-dependent shortest path (SADSP) problem. We show that the SADSP problem is NP-complete. We present inapproximability results and an exact exponential algorithm for this problem. We also extend our results for the longest path problem in arc-dependent networks. Additionally, we explore the problem of detecting negative cycles in arc-dependent networks and discuss its computational complexity. Our results include variants of the negative cycle detection problem such as longest, shortest, heaviest, and lightest negative simple cycles.2

弧相关网络的优化问题分析
本文研究了求解弧相关网络优化问题的算法设计和分析。如果一个电弧A的成本取决于进入A的电弧,则网络被称为电弧依赖网络。这些网络与传统网络有根本不同,传统网络中与电弧相关的成本是一个固定常数,并且是输入的一部分。我们首先研究了arc-dependent最短路径(ADSP)问题,在文献中也称为suffix-1 - path-dependent最短路径问题。如果不要求最短路径是简单的,这个问题有一个多项式时间解。ADSP问题在许多领域都有应用,包括高速公路工程、转弯处罚和禁止以及票价回扣。在本文中,我们感兴趣的是限制在简单路径下的ADSP问题。我们把这个受限的版本称为简单弧相关最短路径问题。我们证明了SADSP问题是np完全的。我们给出了这个问题的不逼近性结果和一个精确的指数算法。我们还扩展了弧相关网络中最长路径问题的结果。此外,我们探讨了在弧相关网络中检测负循环的问题,并讨论了其计算复杂度。我们的结果包括负循环检测问题的变体,如最长、最短、最重和最轻的负简单循环
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来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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