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 T⊆V, 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.