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Approximating Minimum-Cost Connected T-Joins


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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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.

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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).

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