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

本文关注弧相关网络中优化问题的算法设计和分析。如果弧的成本是指网络是弧相关的$\mathrm{一个}$取决于进入的弧线$\mathrm{一个}$. 这些网络与传统网络根本不同，在传统网络中，与弧相关的成本是固定常数并且是输入的一部分。我们首先研究了弧相关最短路径（ADSP）问题，在文献中也称为suffix-1路径相关最短路径问题。如果不要求最短路径简单，则此问题具有多项式时间解。ADSP 问题在许多领域都有应用，包括高速公路工程、转弯处罚和禁令以及票价回扣。在本文中，我们对受限于简单路径的 ADSP 问题感兴趣。我们将此受限版本称为简单的弧相关最短路径 (SADSP) 问题。我们证明了 SADSP 问题是NP 完全的. 我们针对这个问题提出了不可近似的结果和精确的指数算法。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，并讨论了它的计算复杂性。我们的结果包括负循环检测问题的变体，例如最长、最短、最重和最轻的负简单循环。²