Classifying Tractable Instances of the Generalized Cable-Trench Problem

TL;DR

This paper investigates specific tractable cases of the NP-complete generalized cable-trench problem, focusing on graphs with certain structures like edge-disjoint cycles, and provides efficient solution methods for these cases.

Contribution

It identifies graph classes where the generalized cable-trench problem can be solved efficiently and discusses properties affecting its computational complexity.

Findings

Fast solution method for graphs with all cycles edge disjoint

Characterization of graph properties influencing problem tractability

Open questions on broader classes of tractable instances

Abstract

Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma, \tau: E(G) \rightarrow \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge cost with respect to $\tau$ and the single-source shortest paths cost with respect to $\gamma$. Although this problem is provably $NP$-complete in the general case, we examine certain tractable instances involving various graph constructions of trees and cycles, along with quantities associated to edges and vertices that arise out of these constructions. We show that given a graph in which all cycles are edge disjoint, there exists a fast method to determine a cable-trench solution. Further, we examine properties of graphs which contribute to the general intractability of the CTP and present some open questions in this direction.