Graph inference from a walk for trees of bounded degree 3 is NP-complete

Osamu Maruyama, Satoru Miyano

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

Abstract

The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings
EditorsJiri Wiedermann, Petr Hajek
PublisherSpringer Verlag
Pages257-266
Number of pages10
ISBN (Print)3540602461, 9783540602460
Publication statusPublished - Jan 1 1995
Event20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995 - Prague, Czech Republic
Duration: Aug 28 1995Sep 1 1995

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume969
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995
CountryCzech Republic
CityPrague
Period8/28/959/1/95

Fingerprint

Walk
NP-complete problem
Color
Caterpillar
Graph in graph theory
Strings
Hardness
Bound Constraints
Undirected Graph
Vertex of a graph
Class

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Maruyama, O., & Miyano, S. (1995). Graph inference from a walk for trees of bounded degree 3 is NP-complete. In J. Wiedermann, & P. Hajek (Eds.), Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings (pp. 257-266). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 969). Springer Verlag.

Graph inference from a walk for trees of bounded degree 3 is NP-complete. / Maruyama, Osamu; Miyano, Satoru.

Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings. ed. / Jiri Wiedermann; Petr Hajek. Springer Verlag, 1995. p. 257-266 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 969).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Maruyama, O & Miyano, S 1995, Graph inference from a walk for trees of bounded degree 3 is NP-complete. in J Wiedermann & P Hajek (eds), Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 969, Springer Verlag, pp. 257-266, 20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995, Prague, Czech Republic, 8/28/95.
Maruyama O, Miyano S. Graph inference from a walk for trees of bounded degree 3 is NP-complete. In Wiedermann J, Hajek P, editors, Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings. Springer Verlag. 1995. p. 257-266. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Maruyama, Osamu ; Miyano, Satoru. / Graph inference from a walk for trees of bounded degree 3 is NP-complete. Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings. editor / Jiri Wiedermann ; Petr Hajek. Springer Verlag, 1995. pp. 257-266 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{ab4283fc23614a7cbd4f29e707c66155,
title = "Graph inference from a walk for trees of bounded degree 3 is NP-complete",
abstract = "The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.",
author = "Osamu Maruyama and Satoru Miyano",
year = "1995",
month = "1",
day = "1",
language = "English",
isbn = "3540602461",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "257--266",
editor = "Jiri Wiedermann and Petr Hajek",
booktitle = "Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings",
address = "Germany",

}

TY - GEN

T1 - Graph inference from a walk for trees of bounded degree 3 is NP-complete

AU - Maruyama, Osamu

AU - Miyano, Satoru

PY - 1995/1/1

Y1 - 1995/1/1

N2 - The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

AB - The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

UR - http://www.scopus.com/inward/record.url?scp=84947918349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947918349&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84947918349

SN - 3540602461

SN - 9783540602460

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 257

EP - 266

BT - Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings

A2 - Wiedermann, Jiri

A2 - Hajek, Petr

PB - Springer Verlag

ER -