Approximating Prize-Collecting Variants of TSP

TL;DR

An approximation algorithm is presented for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP, and extends to Prize-collecting Multi-Path TSP.

Abstract

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 $\frac{3}{2}+\frac{1}{e}$, 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. By carefully tuning the techniques of the latest results on the aforementioned problems and leveraging the unique properties of our problem, we present a 2.097-approximation algorithm for PCOTSP. A key idea in our result is to first sample a set of trees, and then probabilistically pick up some vertices, while using the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on other vertices to get cheaper parity correction; the sampling probability and the penalty paid by the LP playing a crucial part in both cases. A straightforward adaptation of the aforementioned pruning ideas would only give minuscule improvements over standard parity correction methods. Instead, we use the specific characteristics of our problem together with properties gained from running 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.

Figures (1)

• Figure 1: (a) The graph $T^{\prime\prime}$ after pruning and picking up critical vertices. The terminals in $O$ are drawn as black rectangles. The cycle $C$ is depicted in red, the surviving edges of $R$ are drawn in black and edges in ${F_P}$ in green. The greyed out vertices and edges do not belong to $T"$. They have either been pruned (the dashed vertices and edges), not sampled, or split off. (b) The same graph $T"$ with the various cuts that are considered in the proof of \ref{['lem:t-join-poly']} drawn in different colors. The dashed blue cut is an example for the case where $0 < |S \cap O| < k$. The dashed pink cut shows the case where a pickup edge (drawn in green) is cut. Note that even though the vertex in $S$ was not picked up itself, it is still part of ${F_P}$ and thus must have a high $y$-value. The remaining two cuts (drawn in brown and light blue) show the two cases where $\delta(S)$ contains an odd number of tree edges.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Lemma 1: Splitting off blauth-klein-naegele-2024
• Lemma 2
• Definition 3
• Definition 4
• Lemma 3
• Lemma 4