AI 中文总结
该研究为经典搜索和路由问题引入统一框架，基于折扣因子α和范数参数p，目标是找从根出发使α延迟向量p范数最小的路径，给出p = 1时多项式时间算法及一般p≥1时随机和去随机化算法。
AI 中文摘要
我们介绍了一个用于经典搜索和路由问题的统一框架，涵盖路径搜索、扩展搜索、最小生成树问题和旅行商问题。该框架基于两个参数。一是折扣因子α∈[0,1]，首次遍历边产生全额成本，后续遍历仅产生α倍成本。对于从指定根顶点出发的路径，顶点的α延迟是首次访问该顶点前积累的折扣成本。二是范数参数p≥1。目标是找到一条从根出发访问所有顶点且使α延迟向量的p范数最小的路径。该模型在多个研究充分的目标之间进行插值。对于p = 1和α = 1，恢复路径搜索；对于p = 1和α = 0，恢复扩展搜索。随着p趋于无穷，目标收敛到完工时间类型标准。在端点α = 1和α = 0时，此极限目标分别对应TSP类型和MST类型行为。对于p = 1，我们给出了多项式时间常数因子近似算法，对于一般p≥1，我们获得了具有相同保证的随机常数因子近似算法和去随机化伪多项式时间算法。
英文摘要
We introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor $α\in [0,1]$: the first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an $α$-fraction of this cost. For a path starting at a designated root vertex, the $α$-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter $p\geq 1$. The objective is to find a root-starting path that visits all vertices and minimizes the $p$-norm of the resulting vector of $α$-latencies. The model interpolates between several well-studied objectives. For $p=1$ and $α=1$, it recovers pathwise search; for $p=1$ and $α=0$, it recovers expanding search. As $p$ tends to infinity, the objective converges to a makespan-type criterion. At the endpoints $α=1$ and $α=0$, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For $p=1$, we give polynomial-time constant-factor approximation algorithms for all $α\in[0,1]$, matching the best known guarantees for expanding search at $α=0$ and pathwise search at $α=1$. For general $p\geq 1$, we obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.
