On the maximum TSP with γ-parameterized triangle inequality

Abstract

The maximum TSP with γ-parameterized triangle inequality is defined as follows. Given a complete graph G = (V, E, w) in which the edge weights satisfy w(uv) ≤ γ · (w(ux) + w(xv)) for all distinct nodes ${u,x,v \in V}$, find a tour with maximum weight that visits each node exactly once. Recently, Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) proposed a ${\frac{\gamma+1}{3\gamma}}$-approximation algorithm for ${\gamma\in\left[\frac{1}{2},1\right)}$. In this paper, we show that the approximation ratio of Kostochka and Serdyukov’s algorithm (Upravlyaemye Sistemy 26:55–59, 1985) is ${\frac{4\gamma+1}{6\gamma}}$, and the expected approximation ratio of Hassin and Rubinstein’s randomized algorithm (Inf Process Lett 81(5):247–251, 2002) is ${\frac{3\gamma+\frac{1}{2}}{4\gamma}-O\left(\frac{1}{\sqrt{n}}\right)}$, for ${\gamma\in\left[\frac{1}{2},+\infty\right)}$. These improve the result in Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) and generalize the results in Hassin and Rubinstein and Kostochka and Serdyukov (Inf Process Lett 81(5):247–251, 2002; Upravlyaemye Sistemy 26:55–59, 1985).