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Approximating Maximum-Size Properly Colored Forests

Authors Yuhang Bai, Kristóf Bérczi , Gergely Csáji , Tamás Schwarcz

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Author Details

Yuhang Bai
  • School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi, China
  • Xi'an-Budapest Joint Research Center for Combinatorics, China
Kristóf Bérczi
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary
Gergely Csáji
  • Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary
  • HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary
Tamás Schwarcz
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary

Funding

The research was supported by the Lendület Programme of the Hungarian Academy of Sciences - grant number LP2021-1/2021, by the Ministry of Innovation and Technology of Hungary - grant number ELTE TKP 2021-NKTA-62, and by Dynasnet European Research Council Synergy project - grant number ERC-2018-SYG 810115.
  • Bai, Yuhang: Supported by the National Natural Science Foundation of China - NSFC, grant numbers 12071370 and 12131013 and by Shaanxi Fundamental Science Research Project for Mathematics and Physics - grant number 22JSZ009.
  • Csáji, Gergely: Supported by the Hungarian Scientific Research Fund - grant number K143858, and by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation fund - grant number KDP-2023/C2258525.
  • Schwarcz, Tamás: Supported by the ÚNKP-23-3 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation Fund.

Cite AsGet BibTex

Yuhang Bai, Kristóf Bérczi, Gergely Csáji, and Tamás Schwarcz. Approximating Maximum-Size Properly Colored Forests. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.14

Abstract

In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and (1,2)-Traveling Salesman problems. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithm
  • (1,2)-traveling salesman problem
  • Degree bounded spanning tree
  • Properly colored forest

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