On the relationship between ATSP and the cycle cover problem

Chandran, Sunil L and Ram, Shankar L (2007) On the relationship between ATSP and the cycle cover problem. In: Theoretical Computer Science, 370 (1-3). pp. 218-228.

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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 $c:E=R\rightarrow^+$ is said to satisfy the \gamma-parametrized triangle inequality if \gamma (c(u,w) + c(w, v))\geq c(u, v) for all distinct u, v,w \epsilon V. Then the graph G is called a \gamma-triangular graph. For any \gamma-triangular graph G, for \gamma< 1, we show that $\frac{ATSP(G)}{AP(G)}\leq \frac{\gamma}{1-\gamma}+o(1)$, 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 \gamma-triangular graphs for each valid \gamma< 1 which demonstrates the near-tightness (up to a factor of $\frac {1}{2\gamma}+ o(1))$ of the above bound. For $\gamma \geq 1$, the ratio $\frac {ATSP(G)}{AP(G)}$ 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 \gamma-triangular graph, does there exist a function $f(\gamma)$ such that $\frac{C_{max}}{C_{min}}$ is bounded above by $f(\gamma)?$ (Here $C_{max}$ and $C_{min}$ are the costs of the maximum cost and minimum cost edges respectively.) We show that when $\gamma < \frac{1}{\sqrt 3}$ , $\frac{C_{max}}{C{min}} \leq \frac {2\gamma^3}{1-\gamma^2}$. This upper bound is sharp in the sense that there exist \gamma-triangular graphs with $\frac{C_{max}}{C_{min}} = \frac {2\gamma^3}{1-3 \gamma^2}$ Moreover, for $\gamma \geq \frac {1}{\sqrt 3}$, no such function $f(\gamma)$ exists.

Item Type: Journal Article Theoretical Computer Science Elsevier Copyright of this article belongs to Elsevier. Assignment problem;Cycle cover;Traveling salesman problem;Combinatorial optimization;Approximation algorithms Division of Electrical Sciences > Computer Science & Automation 29 Mar 2007 19 Sep 2010 04:36 http://eprints.iisc.ac.in/id/eprint/10419