On the relationship between ATSP and the cycle cover problem

Abstract

In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, G = ( V , E ) . The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph G = ( V , E ) with a cost function c : E → R + is said to satisfy the γ -parametrized triangle inequality if γ ( c ( u , w ) + c ( w , v ) ) ≥ c ( u , v ) for all distinct u , v , w ∈ V . Then the graph G is called a γ -triangular graph. For any γ -triangular graph G , for γ < 1 , we show that ATSP ( G ) AP ( G ) ≤ γ 1 − γ + o ( 1 ) , where ATSP ( G ) and AP ( G ) are the costs of an optimum Hamiltonian cycle and an optimum cycle cover respectively. In addition, we observe that there exists an infinite family of γ -triangular graphs for each valid γ < 1 which demonstrates the near-tightness (up to a factor of 1 2 γ + o ( 1 ) ) of the above bound. For γ ≥ 1 , the ratio ATSP ( G ) AP ( G ) can become unbounded. The upper bound is shown constructively and can also be viewed as an approximation algorithm for ATSP with parametrized triangle inequality. We also consider the following problem: in a γ -triangular graph, does there exist a function f ( γ ) such that c max c min is bounded above by f ( γ ) ? (Here c max and c min are the costs of the maximum cost and minimum cost edges respectively.) We show that when γ < 1 3 , c max c min ≤ 2 γ 3 1 − 3 γ 2 . This upper bound is sharp in the sense that there exist γ -triangular graphs with c max c min = 2 γ 3 1 − 3 γ 2 . Moreover, for γ ≥ 1 3 , no such function f ( γ ) exists.