Improved Approximation Algorithms for Metric Maximum ATSP and Maximum 3-Cycle Cover Problems

We consider an APX-hard variant (Δ-Max-ATSP) and an APX-hard relaxation (Max-3-DCC) of the classical traveling salesman problem. Δ-Max-ATSP is the following problem: Given an edge-weighted complete loopless directed graph G such that the edge weights fulfill the triangle inequality, find a maximum weight Hamiltonian tour of G. We present a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{31}{40}$\end{document}-approximation algorithm for Δ-Max-ATSP with polynomial running time. Max-3-DCC is the following problem: Given an edge-weighted complete loopless directed graph, compute a spanning collection of node-disjoint cycles, each of length at least three, whose weight is maximum among all such collections. We present a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{3}{4}$\end{document}-approximation algorithm for this problem with polynomial running time. In both cases, we improve on the previous best approximation performances. The results are obtained via a new decomposition technique for the fractional solution of an LP formulation of Max-3-DCC.