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arxiv: 2606.18157 · v1 · pith:5EJQTIV7new · submitted 2026-06-16 · 💻 cs.DS

Reducing Prize-Collecting Stroll and Related Routing Problems to Prize-Collecting TSP

Hong Li This is my paper

Pith reviewed 2026-06-26 21:49 UTC · model grok-4.3

classification 💻 cs.DS
keywords prize-collecting TSPprize-collecting strollapproximation algorithmsrouting problemsmetric graphsreductionΦ-TSP
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The pith

Prize-collecting stroll reduces to prize-collecting TSP, yielding (ρ+ε) approximation when prescribed vertices are constant in number.

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

The paper shows that the prize-collecting-Φ-TSP, a common generalization that includes the prize-collecting stroll, can be reduced to the standard prize-collecting TSP. When the number of prescribed vertices is bounded by a fixed constant, any ρ-approximation for the TSP version lifts to a (ρ+ε)-approximation for the generalized version. The reduction produces a concrete improvement: the prize-collecting stroll now admits an approximation ratio strictly better than 1.6, improving the earlier 1.6662 guarantee.

Core claim

If a ρ-approximation algorithm for the prize-collecting TSP is available, then, for any ε>0, there is a polynomial time (ρ+ε)-approximation algorithm for the prize-collecting-Φ-TSP when the number of prescribed vertices is bounded by a fixed constant.

What carries the argument

The prize-collecting-Φ-TSP, which encodes parity and connectivity requirements on a fixed-size set of prescribed vertices together with the prize-collecting objective on a complete metric graph.

If this is right

  • Any improvement in the approximation ratio for prize-collecting TSP immediately transfers to the prize-collecting stroll and other Φ-constrained variants.
  • The prize-collecting stroll receives an approximation guarantee strictly better than 1.6.
  • Polynomial-time algorithms exist for all prize-collecting-Φ-TSP instances under the constant bound on prescribed vertices.
  • The same reduction applies uniformly to path versions and parity-constrained routing problems that fit inside the Φ model.

Where Pith is reading between the lines

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

  • If future work removes the constant bound on prescribed vertices while preserving the additive ε loss, the result would cover a larger class of routing problems.
  • The technique may combine with dynamic programming over small sets to handle slowly growing numbers of prescribed vertices.
  • Network-design instances that fix parity on a handful of terminals could inherit the same approximation transfer.

Load-bearing premise

The number of prescribed vertices must be bounded by a fixed constant independent of the input size.

What would settle it

A complete metric graph with a constant number of prescribed vertices on which no polynomial-time (ρ+ε)-approximation algorithm exists whenever a ρ-approximation for prize-collecting TSP is given.

read the original abstract

The prize-collecting stroll is the path version of the prize-collecting TSP. Given a complete metric graph, two prescribed terminal vertices $s,t$, and nonnegative penalties on vertices, the prize-collecting stroll asks for an $s$-$t$ tour minimizing the length of the tour plus the total penalty of vertices that are not visited by the $s,t$ tour. We study a common generalization of the prize-collecting stroll and several related prize-collecting routing problems, which we call the prize-collecting-$\Phi$-TSP. In this model, $\Phi$ specifies a set of prescribed vertices alongside their parity and connectivity requirements. We show that, if a $\rho$-approximation algorithm for the prize-collecting TSP is available, then, for any $\varepsilon>0$, there is a polynomial time $(\rho+\varepsilon)$-approximation algorithm for the prize-collecting-$\Phi$-TSP when the number of prescribed vertices is bounded by a fixed constant. This yields a better-than-$1.6$-approximation algorithm for the prize-collecting stroll, improving the previous best-known approximation guarantee of $1.6662$.

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 claims that if a ρ-approximation algorithm exists for the prize-collecting TSP (PC-TSP), then for any ε>0 there is a polynomial-time (ρ+ε)-approximation algorithm for the prize-collecting-Φ-TSP (a generalization that incorporates parity and connectivity requirements on a set Φ of prescribed vertices) whenever |Φ| is a fixed constant and the input is a complete metric graph. This is used to obtain a better-than-1.6 approximation for the prize-collecting stroll, improving the prior 1.6662 guarantee.

Significance. If the reduction holds, the result is significant because it supplies a black-box technique that immediately transfers any future improvement in the PC-TSP approximation ratio to the prize-collecting stroll and several related routing problems with constant-terminal parity/connectivity constraints. The approach follows standard enumeration-over-configurations ideas for constant-terminal problems but is cleanly adapted to the prize-collecting setting with vertex penalties.

minor comments (3)
  1. [Abstract] Abstract: the dependence of the running time on ε (arising from the enumeration over configurations) should be stated explicitly, even if only asymptotically.
  2. [Section 3 (Reduction)] The manuscript should include a short self-contained example (e.g., |Φ|=2 with one parity constraint) illustrating how the modified PC-TSP instance is constructed from the original graph and penalties.
  3. [Theorem 1.1] It would be helpful to state the precise running-time polynomial degree in terms of |Φ| and the input size, so that the result is fully parameterized.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the black-box reduction technique, and the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a polynomial-time reduction showing that any ρ-approximation for prize-collecting TSP yields a (ρ + ε)-approximation for prize-collecting-Φ-TSP (with fixed |Φ|) via enumeration of constant-sized configurations (orders, parities, subsets) followed by a black-box call on a modified instance. This is a standard, non-circular reduction technique for constant-terminal routing problems in metric graphs; the (ρ + ε) loss is explicitly the cost of the reduction step and does not collapse to the input by definition. No self-citations, fitted parameters renamed as predictions, or ansatzes appear in the load-bearing steps. The result is self-contained against external benchmarks (existing PC-TSP approximations).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on standard domain assumptions for metric TSP variants and the existence of a ρ-approximation oracle for PC-TSP; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The input graph is complete and metric.
    Required for the definition of prize-collecting TSP and stroll as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5725 in / 1185 out tokens · 28887 ms · 2026-06-26T21:49:00.196806+00:00 · methodology

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Reference graph

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