Approximability of Connected Factors

TL;DR

This paper studies the complexity and approximation algorithms for finding minimal connected d-factors in graphs, providing new approximation ratios and hardness results related to the TSP and its variants.

Contribution

It introduces approximation algorithms with improved ratios for connected d-factors and extends hardness results, linking these problems to the TSP.

Findings

3-approximation for all d in the undirected case

(r+1)-approximation for even d, where r is TSP ratio

Hardness results matching TSP approximability

Abstract

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard - finding a minimal connected 2-factor is just the traveling salesman problem (TSP). Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected $d$-factor. We give a 3-approximation for all $d$ and improve this to an (r+1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP. Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.