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 T is the set of nodes of odd degree; H may have multiple copies of any edge of G. Recently, An, Kleinberg, and Shmoys (STOC 2012) improved on the long-standing $\frac{5}{3}$ approximation guarantee for the s,t path TSP (the special case where T = {s,t}) and presented an algorithm based on LP rounding that achieves an approximation guarantee of $\frac{1+\sqrt{5}}{2}\approx1.618$. We show that the methods of An et al. extend to the minimum-cost connected T-join problem to give an approximation guarantee of $\displaystyle 5/3 - 1/(9|T|) + O\left(|T|^{-2}\right)$ when |T| ≥ 4; our approximation guarantee is 1.625 when |T| = 4, and it is ≈ 1.642 when |T| = 6. Finally, we focus on a 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 − 2/(|T| − 1) when |T| ≥ 4.