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Approximating Minimum-Cost Connected T-Joins

Abstract

We design and analyse approximation algorithms for the minimum-cost connected T-join problem: given an undirected graph G=(V,E) with nonnegative costs on the edges, and a set of nodes TV, find (if it exists) a spanning connected subgraph H of minimum cost such that every node in T has odd degree and every node not in T has even degree; H may have multiple copies of any edge of G. Two well-known special cases are the TSP (T=∅) and the s,t path TSP (T={s,t}). Recently, An et al. (Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pp. 875–886, 2012) improved on the long-standing \(\frac{5}{3}\) approximation guarantee for the latter problem and presented an algorithm based on LP rounding that achieves an approximation guarantee of \(\frac{1+\sqrt{5}}{2}\approx1.61803\).

We show that the methods of An et al. extend to the minimum-cost connected T-join problem. They presented a new proof for a \(\frac{5}{3}\) approximation guarantee for the s,t path TSP; their proof extends easily to the minimum-cost connected T-join problem. Next, we improve on the approximation guarantee of \(\frac{5}{3}\) by extending their LP-rounding algorithm to get an approximation guarantee of \(\frac{13}{8}=1.625\) for all |T|≥4.

Finally, we focus on the prize-collecting version of the problem, and present a primal-dual algorithm that is “Lagrangian multiplier preserving” and that achieves an approximation guarantee of \(3-\frac{4}{|T|}\) when |T|≥4. Our primal-dual algorithm is a generalization of the known primal-dual 2-approximation for the prize-collecting s,t path TSP. Furthermore, we show that our analysis is tight by presenting instances with |T|≥4 such that the cost of the solution found by the algorithm is exactly \(3-\frac{4}{|T|}\) times the cost of the constructed dual solution.

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Acknowledgements

We thank a number of colleagues for useful discussions; in particular, we thank Jochen Könemann and Chaitanya Swamy. We thank two anonymous reviewers for their comments; these comments resulted in several improvements.

Author information

Correspondence to Joseph Cheriyan.

Additional information

An extended abstract of this work has been published in APPROX–RANDOM 2012, LNCS 7408, pp. 110–121.

J. Cheriyan supported by NSERC grant No. OGP0138432.

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Cheriyan, J., Friggstad, Z. & Gao, Z. Approximating Minimum-Cost Connected T-Joins. Algorithmica 72, 126–147 (2015). https://doi.org/10.1007/s00453-013-9850-8

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Keywords

  • Approximation algorithms
  • LP rounding
  • Primal-dual method
  • Prize-collecting problems
  • T-Joins
  • Traveling Salesman Problem
  • s,t-Path TSP