Special Frequency Quadrilaterals and an Application

Abstract

Given a quadrilateral ABCD in $K_n$ and the distances of edges, the special frequency quadrilaterals are derived as two of the three sum distances $d(A,B)\,+\,d(C,D)$, $d(A,C)\,+\,d(B,D)$, and $d(A,D)\,+\,d(B,C)$ are equal. A probability model formulated based on the special frequency quadrilaterals implies the edges in the optimal Hamiltonian cycle are different from the other edges in $K_n$. Christofides proposed a $\frac{3}{2}$-approximation algorithm for metric traveling salesman problem (TSP) that runs in $O(n^3)$ time. Cornuejols and Nemhauser constructed a family of graphs where the performance ratio of Christofides algorithm is exactly $\frac{3}{2}$ in the worst case. We apply the special frequency quadrilaterals to the family of metric TSP instances for cutting the useless edges. In the end, the complex graph is reduced to a simple graph where the optimal Hamiltonian cycle can be detected in O(n) time, where $n\ge 4$ is the number of vertices in the graph.

The authors acknowledge the funds supported by the Fundamental Research Funds for the Central Universities (No. 2018MS039 and No. 2018ZD09).