[Paper Review] On computing tree and path decompositions with metric constraints on the bags

Guillaume Ducoffe, Sylvain Legay+1 more|arXiv (Cornell University)|Jan 8, 2016

Advanced Graph Theory Research38 references3 citations

TL;DR

This paper establishes the NP-hardness of computing tree-breadth, path-length, path-breadth, and k-good tree decompositions—key metric graph invariants—completing open problems in the literature. It further provides polynomial-time algorithms for recognizing graphs with tree-breadth one in the special cases of planar and bipartite graphs, leveraging structural properties of such graphs and their relation to k-good decompositions.

ABSTRACT

We here investigate on the complexity of computing the \emph{tree-length} and the \emph{tree-breadth} of any graph $G$, that are respectively the best possible upper-bounds on the diameter and the radius of the bags in a tree decomposition of $G$. \emph{Path-length} and \emph{path-breadth} are similarly defined and studied for path decompositions. So far, it was already known that tree-length is NP-hard to compute. We here prove it is also the case for tree-breadth, path-length and path-breadth. Furthermore, we provide a more detailed analysis on the complexity of computing the tree-breadth. In particular, we show that graphs with tree-breadth one are in some sense the hardest instances for the problem of computing the tree-breadth. We give new properties of graphs with tree-breadth one. Then we use these properties in order to recognize in polynomial-time all graphs with tree-breadth one that are planar or bipartite graphs. On the way, we relate tree-breadth with the notion of \emph{$k$-good} tree decompositions (for $k=1$), that have been introduced in former work for routing. As a byproduct of the above relation, we prove that deciding on the existence of a $k$-good tree decomposition is NP-complete (even if $k=1$). All this answers open questions from the literature.

Motivation & Objective

• To resolve the open complexity status of computing tree-breadth, path-length, path-breadth, and k-good tree decompositions.
• To identify tractable graph classes for tree-breadth-one recognition, particularly planar and bipartite graphs.
• To clarify the relationship between k-good tree decompositions and tree-breadth, settling an open question on their computational complexity.
• To explore the border between tractable and intractable instances for metric graph invariants.

Proposed method

• Reduction techniques from known NP-hard problems to prove NP-hardness of tree-breadth, path-length, path-breadth, and k-good tree decompositions.
• Identification and characterization of structural properties of graphs with tree-breadth one, particularly focusing on leaf-vertex types and clique-separators.
• Design of a recursive algorithm for recognizing tree-breadth-one graphs in planar and bipartite graphs, using vertex contraction and edge addition operations.
• Proof by induction on graph size, tracking the invariant 5n - m to show reduction correctness and termination.
• Establishment of a formal link between k-good tree decompositions (for k=1) and tree-breadth, enabling complexity transfer.
• Use of gadget constructions and graph transformations to preserve metric constraints during reductions.

Experimental results

Research questions

• RQ1 Is computing the tree-breadth of a graph NP-hard?
• RQ2 Can graphs with tree-breadth one be recognized in polynomial time for specific graph classes such as planar or bipartite graphs?
• RQ3 What is the computational complexity of deciding the existence of a k-good tree decomposition for k=1?
• RQ4 How do path-length and path-breadth relate to the complexity of minimum distortion embeddings into paths?
• RQ5 Is there a fixed-parameter tractable algorithm for computing metric graph invariants parameterized by clique-number, genus, tree-width, or Hadwiger number?

Key findings

• Computing tree-breadth is NP-hard, resolving an open problem in metric graph theory.
• Path-length and path-breadth are also NP-hard to compute, extending the known NP-hardness of tree-length.
• The problem of deciding the existence of a k-good tree decomposition is NP-complete, even for k=1.
• Graphs with tree-breadth one are the most complex instances for computing tree-breadth, and they admit specific structural properties that enable efficient recognition.
• Polynomial-time recognition algorithms exist for graphs with tree-breadth one in the classes of planar and bipartite graphs.
• The NP-hardness results imply that these metric invariants cannot be approximated below a constant factor, leaving a gap relative to existing constant-factor approximation algorithms.