[Paper Review] A lower bound for the tree-width of planar graphs with vital linkages

Isolde Adler, Philipp Klaus Krause|arXiv (Cornell University)|Nov 9, 2010

Advanced Graph Theory Research13 references4 citations

TL;DR

This paper establishes a single-exponential lower bound on the tree-width of planar graphs with vital linkages, demonstrating that a $(2^k+1)\times(2^k+1)$ grid is insufficient to guarantee irrelevant vertices in the disjoint paths problem. The construction yields a unique solution requiring all vertices and $2^k-1$ crossings of the grid, proving that tree-width and path-width must grow at least as $2^{k-1}+1$ for such graphs.

ABSTRACT

The disjoint paths problem asks, given an graph G and k + 1 pairs of terminals (s_0,t_0), ...,(s_k,t_k), whether there are k+1 pairwise disjoint paths P_0, ...,P_k, such that P_i connects s_i to t_i. Robertson and Seymour have proven that the problem can be solved in polynomial time if k is fixed. Nevertheless, the constants involved are huge, and the algorithm is far from implementable. The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices. We give single exponential lower bounds both for the tree-width of planar graphs with vital linkages, and for the size of the grid necessary for finding irrelevant vertices.

Motivation & Objective

• To establish a lower bound on the tree-width of planar graphs that contain vital linkages, a key component in fixed-parameter tractable algorithms for the disjoint paths problem.
• To demonstrate that the size of the grid required to guarantee irrelevant vertices in planar graphs grows at least exponentially with $k$, challenging the practicality of existing algorithms.
• To construct a planar graph with $k+1$ terminal pairs and a unique solution that uses all vertices and crosses a grid $2^k - 1$ times, proving no vertex is irrelevant.
• To provide a concrete lower bound for the functions $f(k)$ and $g(k)$ in Robertson and Seymour's tree-width and path-width theorems for vital linkages.

Proposed method

• Construct a planar graph $G$ containing a $(2^k+1)\times(2^k+1)$ grid $\mathfrak{G}$, with terminals $s_i$ and $t_i$ placed on the boundary to force increasing numbers of crossings.
• Assign terminals such that $P_0$ crosses the grid 0 times, $P_1$ once, and $P_{i+1}$ crosses twice as often as $P_i$, leading to $k_i = 2^{i-1}$ crossings for $i > 0$.
• Remove regions opposite each terminal pair to enforce that paths must route through the grid and avoid reusing edges, ensuring a unique solution.
• Prove uniqueness of the solution by induction: each path $P_i$ must cross the grid and force subsequent paths to double crossings to avoid blocking.
• Use the grid minor property and known tree-width results: the $n\times n$ grid has tree-width $n$, so a $(2^k+1)\times(2^k+1)$ grid has tree-width $2^k+1$.
• Leverage the fact that the unique solution forms a vital linkage, and apply Robertson and Seymour’s theorem to derive lower bounds on $f(k)$ and $g(k)$.

Experimental results

Research questions

• RQ1 What is the minimum grid size required to guarantee the existence of an irrelevant vertex in planar graphs with $k+1$ terminal pairs?
• RQ2 How large must the tree-width of a planar graph with a vital linkage be, as a function of $k$?
• RQ3 Can a $(2^k+1)\times(2^k+1)$ grid in a planar graph fail to contain an irrelevant vertex for the disjoint paths problem?
• RQ4 What is the tightest lower bound on the functions $f(k)$ and $g(k)$ in Robertson and Seymour’s tree-width and path-width theorems for vital linkages?
• RQ5 Is there a single-exponential lower bound on the parameter dependency in fixed-parameter algorithms for the disjoint paths problem via irrelevant vertex techniques?

Key findings

• A planar graph with $k+1$ terminal pairs exists such that the disjoint paths problem has a unique solution, and no vertex is irrelevant.
• The constructed graph contains a $(2^k+1)\times(2^k+1)$ grid as a minor, and the solution paths cross the grid exactly $2^k - 1$ times.
• The tree-width and path-width of the graph are at least $2^{k-1} + 1$, establishing a lower bound on the functions $f(k)$ and $g(k)$ from Robertson and Seymour’s theorem.
• The $(2^k+1)\times(2^k+1)$ grid is insufficient to guarantee an irrelevant vertex, implying that larger grids are necessary for the irrelevant vertex technique to apply.
• The lower bound on tree-width is tight in the sense that it matches the exponential growth observed in the parameter dependency of fixed-parameter algorithms.
• The result shows that the parameter dependency in the disjoint paths algorithm cannot be improved below single-exponential unless the irrelevant vertex technique is bypassed or refined.