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DOI: 10.4230/LIPIcs.MFCS.2023.77

A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs

Authors: Guozhen Rong, Yongjie Yang, and Wenjun Li

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract

We study the partial search order problem (PSOP) proposed recently by Scheffler [WG 2022]. Given a graph G together with a partial order on the set of vertices of G, this problem determines if there is an 𝒮-ordering that is consistent with the given partial order, where 𝒮 is a graph search paradigm like BFS, DFS, etc. This problem naturally generalizes the end-vertex problem which has received much attention over the past few years. It also generalizes the so-called ℱ-tree recognition problem which has just been studied in the literature recently. Our main contribution is a polynomial-time dynamic programming algorithm for the PSOP of the maximum cardinality search (MCS) restricted to chordal graphs. This resolves one of the most intriguing open questions left in the work of Scheffler [WG 2022]. To obtain our result, we propose the notion of layer structure and study numerous related structural properties which might be of independent interest.

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Guozhen Rong, Yongjie Yang, and Wenjun Li. A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rong_et_al:LIPIcs.MFCS.2023.77,
  author =	{Rong, Guozhen and Yang, Yongjie and Li, Wenjun},
  title =	{{A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{77:1--77:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.77},
  URN =		{urn:nbn:de:0030-drops-186118},
  doi =		{10.4230/LIPIcs.MFCS.2023.77},
  annote =	{Keywords: partial search order, maximum cardinality search, chordal graphs, clique graphs, dynamic programming}
}

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DOI: 10.4230/LIPIcs.ISAAC.2019.1

Graph Searches and Their End Vertices

Authors: Yixin Cao, Zhifeng Wang, Guozhen Rong, and Jianxin Wang

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract

Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al. a decade ago, and only by then we started to realize how little we understand it. Even the apparently naïve question "which vertex can be the last visited by a graph search algorithm," known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of maximum cardinality search. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs, and end vertices of lexicographic depth-first searches on chordal graphs. Finally, we present 2^n * n^O(1)-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, and maximum cardinality searches on general graphs.

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Yixin Cao, Zhifeng Wang, Guozhen Rong, and Jianxin Wang. Graph Searches and Their End Vertices. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cao_et_al:LIPIcs.ISAAC.2019.1,
  author =	{Cao, Yixin and Wang, Zhifeng and Rong, Guozhen and Wang, Jianxin},
  title =	{{Graph Searches and Their End Vertices}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{1:1--1:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.1},
  URN =		{urn:nbn:de:0030-drops-114973},
  doi =		{10.4230/LIPIcs.ISAAC.2019.1},
  annote =	{Keywords: maximum cardinality search, (lexicographic) breadth-first search, (lexicographic) depth-first search, chordal graph, weighted clique graph, end vertex}
}

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Documents authored by Rong, Guozhen

A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs

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Graph Searches and Their End Vertices

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