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All Colors Shortest Path Problem

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Yunus Can Bilge, Doğukan Çağatay, Begüm Genç, Mecit Sar{i}, Hüseyin Akcan, Cem Evrendilek

Computational Complexity · Submitted 24 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we prove that All Colors Shortest Path problem is NP-hard, and does not lend itself to a constant factor approximation.

how it is stated · asserted 1.00 read the record →

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arXiv ID1507.06865 (version 1)
Pith IDPW24FTBI, Pith's permanent record of this paper and its review
CitationBeg\"um Gen\c{c}, Cem Evrendilek, Do\u{g}ukan \c{C}a\u{g}atay, H\"useyin Akcan, Mecit Sar{\i}, Yunus Can Bilge. "All Colors Shortest Path Problem." arXiv:1507.06865 (2015). https://pith.science/paper/PW24FTBI
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,626 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.98 · states limitation 0.74 · human participants 0.02 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

All Colors Shortest Path problem defined on an undirected graph aims at finding a shortest, possibly non-simple, path where every color occurs at least once, assuming that each vertex in the graph is associated with a color known in advance. To the best of our knowledge, this paper is the first to define and investigate this problem. Even though the problem is computationally similar to generalized minimum spanning tree, and the generalized traveling salesman problems, allowing for non-simple paths where a node may be visited multiple times makes All Colors Shortest Path problem novel and computationally unique. In this paper we prove that All Colors Shortest Path problem is NP-hard, and does not lend itself to a constant factor approximation. We also propose several heuristic solutions for this problem based on LP-relaxation, simulated annealing, ant colony optimization, and genetic algorithm, and provide extensive simulations for a comparative analysis of them. The heuristics presented are not the standard implementations of the well known heuristic algorithms, but rather sophisticated models tailored for the problem in hand. This fact is acknowledged by the very promising results reported.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 All Colors Shortest Path problem defined on an undirected graph aims at finding a shortest, possibly non-simple, path where every color occurs at least once, assuming that each vertex in the graph is associated with a color known in advance. background 0.99 0.17 0.04 asserted 0.99
s1 To the best of our knowledge, this paper is the first to define and investigate this problem. gap 0.62 0.52 0.04 asserted 0.85
s2 Even though the problem is computationally similar to generalized minimum spanning tree, and the generalized traveling salesman problems, allowing for non-simple paths where a node may be visited multiple times makes All Colors Shortest Path problem novel and computationally unique. background 0.47 0.41 0.05 asserted 1.00
s3 In this paper we prove that All Colors Shortest Path problem is NP-hard, and does not lend itself to a constant factor approximation. result 0.97 0.96 0.64 asserted 1.00
s4 We also propose several heuristic solutions for this problem based on LP-relaxation, simulated annealing, ant colony optimization, and genetic algorithm, and provide extensive simulations for a comparative analysis of them. contribution 0.70 0.96 0.07 asserted 1.00
s5 The heuristics presented are not the standard implementations of the well known heuristic algorithms, but rather sophisticated models tailored for the problem in hand. method 0.50 0.83 0.03 asserted 1.00
s6 This fact is acknowledged by the very promising results reported. interpretation 0.69 0.70 0.08 asserted 0.88

What kind of paper the classifier reports this is: method or tool 0.56 · evidence from simulation or computation 0.68.

What was examined, and what was not

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  • references and citing sentences
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

All Colors Shortest Path problem defined on an undirected graph aims at finding a shortest, possibly non-simple, path where every color occurs at least once, assuming that each vertex in the graph is associated with a color known in advance. To the best of our knowledge, this paper is the first to define and investigate this problem. Even though the problem is computationally similar to generalized minimum spanning tree, and the generalized traveling salesman problems, allowing for non-simple paths where a node may be visited multiple times makes All Colors Shortest Path problem novel and computationally unique. In this paper we prove that All Colors Shortest Path problem is NP-hard, and does not lend itself to a constant factor approximation. We also propose several heuristic solutions for this problem based on LP-relaxation, simulated annealing, ant colony optimization, and genetic algorithm, and provide extensive simulations for a comparative analysis of them. The heuristics presented are not the standard implementations of the well known heuristic algorithms, but rather sophisticated models tailored for the problem in hand. This fact is acknowledged by the very promising results reported.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper
  • The All Colors Shortest Path problem (ACSP) [ref] asks the shortest path in an undirected graph with nodes colored, under the constraint that the paths must visit all colors.

    Quick Brown Fox in Formal Languages · Related Work · 0.97

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