Classifying Tractable Instances of the Generalized Cable-Trench Problem

TL;DR

The paper tackles the generalized cable-trench problem, combining MST costs with respect to one edge-weight and SPT costs with respect to another, and proves NP-hardness in general while developing tractable frameworks for structured graphs. It introduces the wedge operation and leverages cycle and theta-graph decompositions to propagate local cable-trench solutions into global ones, aided by the notions of strength indices and breaking edges. Specifically, cactus graphs and theta-graphs are shown to be tractable via precomputation of strengths and systematic wedging, enabling polynomial-time assembly of global solutions. The work highlights the delicate balance between graph structure and tractability, discusses inductive strategies for broader classes, and identifies inherent limits where exponential spanning-tree counts impede efficient solution retrieval, providing open questions for extending strength-based methods.

Abstract

Given a graph $G$ rooted at a vertex $r$ and weight functions, $γ, τ: E(G) \rightarrow \mathbb{R}$, 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 $τ$ and the single-source shortest paths cost with respect to $γ$. 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.

Figures (6)

• Figure 1: Cabling $B$ through $A$ requires laying $k$ cables of length $\sum_{i=1}^n e_i$ to reach the endpoint of $A$, as well as the additional cabling cost of path $B$. The cabling cost of concatenating paths $A$ and $B$ requires pre-cabling of path $B$ through path $A$.
• Figure 2: The cycle $G$ and distinguished edge $e$ with labeled components $P, Q,$ and $R$.
• Figure 3: Wedging multiple graphs onto non-root vertices on the cycle $G$.
• Figure 4: The $\theta$-graph $G$ with distinguished edges $e_1, e_2$ along with labeled components for the corresponding decomposition.
• Figure 5: Illustrative example of the biconnected component decomposition (or 'block-cut tree' decomposition of a cactus graph into cycles $A, D, F$ and trees $B, C, E$.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Definition 2.7
• Definition 2.8
• Proposition 3.1