[Paper Review] Tree spanners of small diameter

Ioannis Papoutsakis|arXiv (Cornell University)|Mar 20, 2015

Computational Geometry and Mesh Generation18 references3 citations

TL;DR

This paper investigates the existence of tree t-spanners with diameter at most t+1 in undirected graphs. It presents a polynomial-time algorithm for t=3 and proves NP-completeness for t≥4, resolving the complexity status for t≤3 and establishing hardness for larger t.

ABSTRACT

A graph that contains a spanning tree of diameter at most $t$ clearly admits a tree $t$-spanner, since a tree $t$-spanner of a graph $G$ is a sub tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. In this paper, graphs that admit a tree $t$-spanner of diameter at most $t+1$ are studied. For $t$ equal to 1 or 2 the problem has been solved. For $t=3$ we present an algorithm that determines if a graph admits a tree 3-spanner of diameter at most 4. For $t\geq4$ it is proved that it is an NP-complete problem to decide whether a graph admits a tree $t$-spanner of diameter at most $t+1$.

Motivation & Objective

• To determine the computational complexity of deciding whether a graph admits a tree t-spanner of diameter at most t+1.
• To resolve the open complexity status of the tree 3-spanner problem with respect to diameter constraints.
• To characterize graph families that admit such spanners and provide algorithmic solutions for small t.
• To establish a dichotomy between tractable and NP-complete cases based on the value of t.

Proposed method

• Develops a polynomial-time algorithm to test for the existence of a tree 3-spanner with diameter at most 4.
• Uses the concept of t-centers (a central vertex or edge) to define t-star graphs and relate them to spanner existence.
• Applies breadth-first search trees rooted at t-centers to construct candidate spanners with bounded diameter.
• Employs structural graph theory, including properties of induced cycles and perfect graphs, to analyze spanner unions.
• Reduces known NP-complete problems to prove NP-completeness for t≥4 using graph constructions like h(f(I),t).
• Uses the equivalence between t-star graphs and graphs admitting spanning trees of diameter at most t (Lemma 1).

Experimental results

Research questions

• RQ1 For which values of t is the problem of deciding whether a graph admits a tree t-spanner of diameter at most t+1 tractable?
• RQ2 Can an efficient algorithm be designed to recognize graphs admitting a tree 3-spanner of diameter at most 4?
• RQ3 What structural properties of graphs ensure the existence of such spanners with small diameter?
• RQ4 Is the problem NP-complete for all t≥4, and if so, under what conditions?
• RQ5 How do the unions of tree t-spanners behave, particularly in terms of induced cycle structure?

Key findings

• The problem of deciding whether a graph admits a tree t-spanner of diameter at most t+1 is solvable in polynomial time for t≤3.
• A specific polynomial-time algorithm is provided for the case t=3, determining whether a graph admits a tree 3-spanner of diameter at most 4.
• For t≥4, the problem is proven to be NP-complete, establishing a dichotomy in complexity.
• The paper shows that the union of any two tree 3-spanners cannot contain an odd induced cycle other than a triangle, and such unions are perfect graphs.
• Graphs that admit a t-center (t-star graphs) are exactly those that admit a spanning tree of diameter at most t.
• The construction of a tree t-spanner from a shortest-path tree rooted at a t-center ensures diameter at most t+1 for t≥4, but the decision problem remains NP-complete.