k-Trails: Recognition, Complexity, and Approximations

TL;DR

This paper investigates the computational properties of k-trails, a class of degree-constrained spanning hierarchies, establishing recognition algorithms, complexity results, and approximation methods using matroid theory.

Contribution

It introduces a matroid-based framework for analyzing k-trails, resolving open questions on recognition, existence, and approximation of minimum weight k-trails.

Findings

Recognition of k-trails can be done efficiently.

Deciding if a graph contains a k-trail is NP-complete.

A (2k-1)-trail approximation can be found with bounded weight.

Abstract

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo.