## Youngho Yoo gave a talk on the approximation algorithm for the traveling salesman problem on cubic graphs at the Discrete Math Seminar

On January 3, 2023, Youngho Yoo from Texas A&M University gave a talk at the Discrete Math Seminar on the approximation algorithm for the traveling salesman problem on cubic graphs by showing an upper bound on the length of a shortest closed spanning walk in a simple 2-connected subcubic graph. The title of his talk was “Approximating TSP walks in subcubic graphs“.

## Youngho Yoo (유영호), Approximating TSP walks in subcubic graphs

The Graphic Travelling Salesman Problem is the problem of finding a spanning closed walk (a TSP walk) of minimum length in a given connected graph. The special case of the Graphic TSP on subcubic graphs has been studied extensively due to their worst-case behaviour in the famous $\frac{4}{3}$-integrality-gap conjecture on the “subtour elimination” linear programming relaxation of the Metric TSP.

We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král’, and Mohar. This bound is best possible and we characterize the extremal subcubic examples meeting this bound. We also give a quadratic time combinatorial algorithm to find such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the Graphic TSP on cubic graphs. Joint work with Michael Wigal and Xingxing Yu.

기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209