A97-approximation algorithm for Graphic TSP in cubic bipartite graphs

Jeremy A. Karp, R. Ravi
2016 Discrete Applied Mathematics  
We prove new results for approximating the Graphic TSP. Specifically, we provide a polynomial-time 9 7approximation algorithm for cubic bipartite graphs and a ( 9 7 + 1 21(k−2) )-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with
more » ... ur knowledge of the structure of cubic graphs. for instances of sub-quartic graphs [13] . In a recent preprint, van Zuylen [15] extends the work in [5, 12] to obtain a 5 4 -approximation algorithm for cubic bipartite graphs and a ( 4 3 − 1 8754 )-approximation algorithm for 2-connected cubic graphs. Progress in approximating the Graphic TSP in cubic graphs also relates to traditional graph theory, as Barnette's conjecture [2] states that all bipartite, planar, 3-connected, cubic graphs are Hamiltonian. This conjecture suggests that instances of Graph TSP on Barnette graphs could be easier to approximate, and conversely, approximation algorithms for the Graphic TSP in Barnette graphs may lead to the resolution of this conjecture. Indeed, Correa, Larré, and Soto [5] provided a ( 4 3 − 1 18 )-approximation algorithm for Barnette graphs. Along these lines, Aggarwal, Garg, and Gupta [1] were able to obtain a 4 3 -approximation algorithm for 3-edge-connected cubic graphs before any 4 3 -approximation algorithms were known for all cubic graphs. In this paper, we examined graphs that are cubic and bipartite, another class of graphs that includes all Barnette graphs. An improved approximation for this class of graphs is the primary theoretical contribution of this paper. Theorem 1.1. Given a cubic bipartite connected graph G with n vertices, there is a polynomial time algorithm that computes a spanning Eulerian multigraph H in G with at most 9 7 n edges. Corollary 1.2. Given a k-regular bipartite connected graph G with n vertices where k ≥ 4, there is a polynomial time algorithm that computes a spanning Eulerian multigraph H in G with at most ( 9 7 + 1 21(k−2) )n−2 edges. This extension complements results [16, 7] which provide guarantees for k-regular graphs in the asymptotic regime. Corollary 1.2 improves on these guarantees for small values of k. Note that even for k = 4 Corollary 1.2 yields a solution with fewer then 4 3 n edges.
doi:10.1016/j.dam.2015.10.038 fatcat:kydydtum6nbbhljrxzolvhpbwa