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An improved approximation algorithm for the minimum 3-path partition problem

Author

Listed:
  • Yong Chen

    (Hangzhou Dianzi University)

  • Randy Goebel

    (University of Alberta)

  • Guohui Lin

    (University of Alberta)

  • Bing Su

    (Xi’an Technological University)

  • Yao Xu

    (University of Alberta)

  • An Zhang

    (Hangzhou Dianzi University)

Abstract

Given a graph $$G = (V, E)$$ G = ( V , E ) , we seek for a collection of vertex disjoint paths each of order at most 3 that together cover all the vertices of V. The problem is called 3-path partition, and it has close relationships to the well-known path cover problem and the set cover problem. The general k-path partition problem for a constant $$k \ge 3$$ k ≥ 3 is NP-hard, and it admits a trivial k-approximation. When $$k = 3$$ k = 3 , the previous best approximation ratio is 1.5 due to Monnot and Toulouse (Oper Res Lett 35:677–684, 2007), based on two maximum matchings. In this paper we first show how to compute in polynomial time a 3-path partition with the least 1-paths, and then apply a greedy approach to merge three 2-paths into two 3-paths whenever possible. Through an amortized analysis, we prove that the proposed algorithm is a 13 / 9-approximation. We also show that the performance ratio 13 / 9 is tight for our algorithm.

Suggested Citation

  • Yong Chen & Randy Goebel & Guohui Lin & Bing Su & Yao Xu & An Zhang, 2019. "An improved approximation algorithm for the minimum 3-path partition problem," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 150-164, July.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:1:d:10.1007_s10878-018-00372-z
    DOI: 10.1007/s10878-018-00372-z
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    Cited by:

    1. Shiming Li & Wei Yu & Zhaohui Liu, 2024. "Improved approximation algorithms for the k-path partition problem," Journal of Global Optimization, Springer, vol. 90(4), pages 983-1006, December.

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