Jingyang Zhao, Jason William Setiawan, Uman Khalid, Chaoning Zhang, Een-Kee Hong, Hyundong Shin
原文摘要
Sweep coverage requires each point of interest (PoI) to be revisited by some mobile sensor at least once during every sweep period, and it underlies persistent patrolling, environmental monitoring, and infrastructure inspection. Given a set of $N$ PoIs, each associated with a nonnegative weight representing its importance, the weighted budgeted sweep coverage problem (WB-SCP) seeks routing trajectories for a budget of $M\leq N$ identical mobile sensors with a common maximum speed to maximize the total weight of the sweep-covered PoIs. For the unweighted special case in which every PoI has unit weight, the first constant-factor approximation algorithm achieves a ratio of $0.0116-O(\varepsilon)$ for any constant $\varepsilon>0$, and whether the WB-SCP itself admits such an algorithm has remained open. In this paper, we resolve this open question by proposing two constant-factor approximation algorithms for the WB-SCP. The first converts any bifactor $(α,β)$-approximation algorithm into a $β/(2α)$-approximation algorithm, yielding a ratio of $1/16=0.0625$. The second builds on a greedy maximum-coverage framework and improves the ratio to $1-e^{-1/2}-\varepsilon$ for any fixed $\varepsilon>0$. Both algorithms also improve the best-known ratio for the unweighted case, and the second extends to heterogeneous sensors with different speeds while preserving the same guarantee.
Jingyang Zhao, Jason William Setiawan, Uman Khalid, Chaoning Zhang, Een-Kee Hong, Hyundong Shin
原文摘要
The metric traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization that asks for a minimum-cost Hamiltonian cycle in a complete undirected graph with $N$ vertices, where edge costs satisfy the triangle inequality. The biased TSP (BTSP) is a natural generalization of the metric TSP, in which the vertex set is partitioned into $k$ subsets: edge costs satisfy the triangle inequality within each subset, while edges crossing the subsets may violate the triangle inequality by a certain bias factor. This model captures scenarios in which distances are metric only locally but may be distorted across different regions or types of vertices. In this paper, we study approximation algorithms for the BTSP and its path variant, the biased traveling salesman path problem (BTSPP), which asks for a minimum-cost Hamiltonian path. Previously, constant-factor approximation algorithms were known only for the case $k=2$. Whether the BTSP and the BTSPP admit constant-factor approximation algorithms for any fixed integer $k>2$ has remained an open question. We resolve this question affirmatively. We show that both the BTSP and the BTSPP admit $O(k)$-approximation algorithms with running time $N^{O(2^k k!)}$. Moreover, for the BTSPP with $k=2$, we propose an improved $(α+2+\varepsilon)$-approximation algorithm, where $α<3/2-10^{-34}$ is the best-known approximation ratio for the metric TSP, improving upon the previous ratio of $4$