Open Access

Morteza Alimi, Tobias Mömke, Michael Ruderer

• We present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below 1.6, shown by Blauth, Klein and Nägele [IPCO 2024]. The best approximation ratio for Ordered TSP is 3/2+1/e, presented by Böhm, Friggstad, Mömke, Spoerhase [SODA 2025] and Armbruster, Mnich, Nägele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP. We present a 2.097-approximation algorithm for PCOTSP, which is, to the best of our knowledge, the first result for this problem. Key ideas in our approach are to sample a set of trees and then to probabilistically pick up some vertices, and to use the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on the sampled vertices. While the sampling probability of vertices for our problem isWe present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below 1.6, shown by Blauth, Klein and Nägele [IPCO 2024]. The best approximation ratio for Ordered TSP is 3/2+1/e, presented by Böhm, Friggstad, Mömke, Spoerhase [SODA 2025] and Armbruster, Mnich, Nägele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP. We present a 2.097-approximation algorithm for PCOTSP, which is, to the best of our knowledge, the first result for this problem. Key ideas in our approach are to sample a set of trees and then to probabilistically pick up some vertices, and to use the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on the sampled vertices. While the sampling probability of vertices for our problem is lower than for PCTSP, intuitively leaving less spare penalty to spend, we leverage the cycle structure induced by the sampled trees together with a simple combinatorial algorithm to bring the approximation factor below 2.1. Our techniques extend to Prize-collecting Multi-Path TSP, building on results from Böhm, Friggstad, Mömke, Spoerhase [SODA 2025], leading to a 2.41-approximation.…