[Paper Review] A 9/7-Approximation Algorithm for Graphic TSP in Cubic Bipartite Graphs

Jeremy Karp, R. Ravi|arXiv (Cornell University)|Nov 14, 2013

Advanced Graph Theory Research14 references3 citations

TL;DR

This paper presents a polynomial-time 9/7-approximation algorithm for the Graphic TSP in cubic bipartite graphs by constructing a spanning Eulerian multigraph with at most 9/7n edges. The approach uses cycle compression and expansion techniques to minimize the number of cycles in a 2-factor, leveraging the even-cycle structure of bipartite graphs and Petersen’s theorem to find edge-disjoint perfect matchings.

ABSTRACT

We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time \frac{9}{7}-approximation algorithm for cubic bipartite graphs and a (\frac{9}{7}+\frac{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 our knowledge of the structure of cubic graphs.

Motivation & Objective

• To improve the approximation ratio for Graphic TSP in cubic bipartite graphs beyond previous bounds.
• To address the challenge of minimizing the number of cycles in a 2-factor to reduce the size of the resulting Eulerian multigraph.
• To extend the algorithm to k-regular bipartite graphs and achieve a tighter approximation factor for small k.
• To provide theoretical progress toward resolving Barnette’s conjecture by studying Hamiltonian properties in structured cubic graphs.
• To demonstrate that Graphic TSP instances in cubic bipartite graphs can be approximated more effectively than general Metric TSP.

Proposed method

• Uses Petersen’s theorem to decompose a cubic bipartite graph into three edge-disjoint perfect matchings.
• Constructs a 2-factor by combining two of the three perfect matchings, forming a cycle cover.
• Applies a compression and expansion process to reduce the number of cycles in the 2-factor by contracting 4-cycles and handling 6-cycles via iterative contraction.
• Employs a modified version of the DOUBLETREE procedure to extend the 2-factor into a spanning Eulerian multigraph using 2(k−1) additional edges.
• For k-regular bipartite graphs, selects a cubic subgraph with few K3,3 components using a matching-based subroutine to minimize cycle count.
• Uses the pigeonhole principle to bound the number of K3,3 components in the selected cubic subgraph, enabling a refined approximation factor.

Experimental results

Research questions

• RQ1 Can a better approximation ratio than 3/2 be achieved for Graphic TSP in cubic bipartite graphs?
• RQ2 How can cycle compression and expansion techniques be used to minimize the number of cycles in a 2-factor of a cubic bipartite graph?
• RQ3 What is the tightest possible approximation factor for Graphic TSP in k-regular bipartite graphs for small k ≥4?
• RQ4 Can the number of K3,3 components in a cubic subgraph of a k-regular bipartite graph be bounded to improve the approximation ratio?
• RQ5 Does the structure of cubic bipartite graphs allow for a better approximation than known for general Graphic TSP instances?

Key findings

• The paper achieves a 9/7-approximation for Graphic TSP in cubic bipartite graphs, improving upon previous results.
• The algorithm computes a spanning Eulerian multigraph with at most 9/7n edges in O(n^5/2) time.
• For k-regular bipartite graphs with k ≥4, the algorithm achieves a (9/7 + 1/(21(k−2)))-approximation, which is tighter than prior bounds for small k.
• The number of K3,3 components in the cubic subgraph is bounded by n/(6(k−2)), enabling the improved approximation factor.
• The algorithm ensures that the 2-factor has at most n/7 cycles in the compressed graph, which is critical for minimizing the final multigraph size.
• The method proves effective even for k=4, yielding a solution with fewer than 4/3n edges, demonstrating practical improvement.