Problem
Research questions and friction points this paper is trying to address.
Approximating Prize-Collecting TSP tour cost with penalties
Improving approximation ratio below 1.6 for PC-TSP
Developing polynomial-time algorithm for PC-TSP and stroll
A Better-Than-1.6-Approximation for Prize-Collecting TSP
🤖 AI Summary
This paper studies the Prize-Collecting Traveling Salesman Problem (PCTSP), which jointly optimizes tour length and penalty costs for unvisited vertices. To overcome the long-standing 1.774 approximation barrier, we propose a novel analytical framework integrating tree decomposition, structured pruning, Eulerian augmentation, and parity correction, augmented with refined probabilistic analysis and the golden ratio bounding technique. Our approach achieves the first polynomial-time approximation ratio of 1.599 for PCTSP—breaking the theoretical 1.6 threshold—and improves the approximation ratio for the Prize-Collecting Stroll from 1.926 to 1.6662. Both results constitute the current state-of-the-art, representing a significant advance in the approximability of PCTSP and its variants.
📝 Abstract
Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below $1.6$, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of $1.774$. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by $(1+sqrt{5})/2 approx 1.618$, the golden ratio. With some additional technical care we further improve it to $1.599$. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Uses LP relaxation decomposition into rooted trees
Applies pruning step before parity correction
Achieves approximation ratio below 1.6
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