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arxiv: 2605.24944 · v1 · pith:BRQ5EONEnew · submitted 2026-05-24 · 💻 cs.DS

Approximation algorithms for the prize-collecting rural postman problem

Hong Li , Jianping Li , Wei Li , Runtao Xie , Xiaoxiao Yang This is my paper

Pith reviewed 2026-06-30 00:01 UTC · model grok-4.3

classification 💻 cs.DS
keywords prize-collecting rural postman problemapproximation algorithmsrural postman problemprize-collecting traveling salesman problemgraph theorycombinatorial optimization
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The pith

A polynomial-time approximation algorithm for the prize-collecting rural postman problem achieves a ratio strictly smaller than 1.6.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that reducing the prize-collecting rural postman problem to the prize-collecting traveling salesman problem cannot yield an approximation ratio better than two. It then presents a new polynomial-time algorithm that achieves a ratio strictly less than 1.6 for the original problem. This improves on the inherent barrier of the reduction approach. The algorithm is tested on a public benchmark with 118 instances, yielding average and maximum optimality gaps of 3.39% and 12.12%.

Core claim

The authors prove the existence of a polynomial-time approximation algorithm for PCRPP with approximation ratio strictly smaller than 1.6 by means of a direct algorithmic construction that does not rely on solving a PCTSP instance.

What carries the argument

A direct algorithmic construction for PCRPP instances that produces solutions with objective value less than 1.6 times the optimum.

If this is right

  • The PCRPP admits better approximation than the factor-two limit imposed by PCTSP reductions.
  • Practical solutions for PCRPP on graphs with edge lengths and profits can be computed efficiently with small optimality gaps.
  • Similar problems in postman and routing with prizes may benefit from direct constructions rather than reductions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ratio can be improved further, it might approach the best known ratios for related problems like the rural postman problem.
  • The benchmark performance suggests the algorithm could be useful in applications like network maintenance with optional edges.

Load-bearing premise

The central claim depends on the existence and correctness of a specific algorithmic construction whose ratio analysis stays below 1.6.

What would settle it

Finding a PCRPP instance where the new algorithm produces a solution whose cost is at least 1.6 times the optimal cost would falsify the ratio claim.

Figures

Figures reproduced from arXiv: 2605.24944 by Hong Li, Jianping Li, Runtao Xie, Wei Li, Xiaoxiao Yang.

Figure 1
Figure 1. Figure 1: A factor-two barrier for the natural reduction-based approach from the PCRPP to the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the edge-profit core. Solid edges are positive-profit edges, dashed edges [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

In this paper, we study the prize-collecting rural postman problem (PCRPP), a variant of the rural postman problem. Given a PCRPP instance consisting of an undirected graph whose edges have nonnegative lengths and nonnegative profits, together with a root vertex, the goal is to find a closed walk that starts and ends at the root vertex and minimizes the sum of its length and the profits of all edges that the walk does not traverse. A natural way to design an approximation algorithm for the PCRPP is to construct a prize-collecting traveling salesman problem (PCTSP) instance from the given PCRPP instance, apply an approximation algorithm to the PCTSP instance, and then convert the resulting PCTSP solution into a PCRPP solution. We show that this approach has an inherent factor-two barrier: even if the constructed PCTSP instance is solved exactly, the resulting PCRPP solution can have objective value arbitrarily close to twice the optimum value of the original PCRPP instance. Our main result is a polynomial-time approximation algorithm with approximation ratio strictly smaller than 1.6 for the PCRPP. On a public 118-instance benchmark set, the proposed algorithm has average and maximum optimality gaps of 3.39\% and 12.12\%, respectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the prize-collecting rural postman problem (PCRPP) on undirected graphs with edge lengths and profits. It proves that any reduction to the prize-collecting TSP yields solutions whose PCRPP objective can be arbitrarily close to twice the optimum, even when the PCTSP instance is solved exactly. The central contribution is a distinct polynomial-time algorithm, constructed via prize-collecting matching followed by tree-doubling steps, whose charging argument establishes an approximation ratio strictly less than 1.6. On a 118-instance public benchmark the algorithm reports average and maximum optimality gaps of 3.39% and 12.12%.

Significance. The explicit factor-2 barrier clarifies a structural limitation of the natural reduction and justifies the need for a bespoke construction. The new ratio below 1.6 is a concrete improvement for a classic routing problem; the charging argument that splits the optimum into traversed and untraversed components appears self-contained and may extend to related prize-collecting problems. The reported benchmark gaps indicate that the theoretical guarantee is accompanied by practical performance.

minor comments (3)
  1. [Abstract] The exact numerical ratio achieved by the new algorithm (beyond the strict inequality <1.6) is not stated in the abstract or introduction; stating the concrete bound would make the improvement easier to compare with prior work.
  2. The description of the 118-instance benchmark set does not specify the source or generation procedure of the instances; adding a reference or brief generation summary would improve reproducibility.
  3. Notation for the prize-collecting matching subroutine and the subsequent tree-doubling step could be introduced with a short table of symbols to reduce cross-referencing in the analysis section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the recognition of the factor-2 barrier result, and the recommendation of minor revision. The referee's description of the manuscript is accurate.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a new algorithmic construction for PCRPP (via prize-collecting matching and tree-doubling) whose <1.6 ratio is proved by a direct charging argument that bounds the returned walk against an optimal solution by splitting into traversed and untraversed components. This analysis is independent of the PCTSP reduction (which the paper separately shows has an inherent 2-barrier via an explicit family of instances) and contains no fitted parameters, self-definitional equations, or load-bearing self-citations. The derivation is self-contained and externally falsifiable on the 118-instance benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard undirected-graph assumptions and approximation-algorithm techniques without introducing fitted parameters or new postulated entities.

axioms (1)
  • standard math Undirected graphs with nonnegative edge lengths and nonnegative profits admit the stated problem definition and algorithmic constructions.
    The PCRPP definition and all claims presuppose these standard graph properties.

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Reference graph

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19 extracted references · 18 canonical work pages · 2 internal anchors

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