Forbidden Minors And Hidden Algorithms: Navigating Non-Constructivity In Graph Theory

Deciphering Non-Constructivity in Graph Minor Theory

What are Graph Minors?

A graph minor of a graph G is a graph that can be obtained from G by contracting edges, deleting edges and deleting isolated vertices. More formally, a graph H is called a minor of a graph G if H can be formed from G by contracting edges, deleting edges, and deleting isolated vertices. Some key properties of minor relationships include:

Minor relationships demonstrate subgraph relationships – if H is a minor of G, then H is also a subgraph of G
Minor relationships are transitive – if H is a minor of G, and J is a minor of H, then J is also a minor of G
Testing for the existence of a minor can be done in polynomial time using algorithms, but finding the actual minor subgraph is often computationally Hard

Some examples of minor relationships include:

Any tree is a minor of a complete graph or grid graph
A cycle graph Cn is a minor of a complete bipartite graph Km,n where m, n >= 3
Planar graphs are minors of graphs embeddable on a torus or Klein bottle

Kuratowski’s Theorem – Early Connections to Non-Constructivity

In 1930, Kazimierz Kuratowski provided an early forbidden minor characterization of planar graphs. Specifically, he proved that a graph is planar if and only if it does not contain a K5 minor or K3,3 minor. While an existence proof, Kuratowski’s theorem does not actually constructively identify where the forbidden minors may be hiding in a non-planar graph.

Table of Contents

Kuratowski used the topological fact that planar graphs can be drawn on a plane without any edge crossings. By contrast, the complete graph K5 and the complete bipartite graph K3,3 are non-planar as they intrinsically require edges to cross when embedded on a plane. However, Kuratowski relied on non-constructive proofs about existence of minors rather than providing an algorithm for extracting the minors.

Robertson and Seymour’s Graph Minor Project

Overview of the Graph Minor Theorem

In a sweeping 20 paper project spanning decades, Robertson and Seymour fundamentally characterized all possible minor relationships in their Graph Minor Theorem. Specifically, they proved that for any infinite sequence G1, G2, G3… of graphs, there exist indices i < j such that Gi is a minor of Gj. Intuitively, this means any infinite set of graphs must contain a pair with a minor relationship.

To enable this result, Robertson and Seymour developed an enormous mathematical machinery around graph minors, including structural decompositions, reductions, and OBSTRUCTION sets that characterize planarity, linkages, and more. However, their proofs remain finely tuned to an existential viewpoint rather than explicitly yielding algorithms.

Inherently Non-Constructive Nature

While the Graph Minor Theorem reveals deep structure about the anatomy of all possible graph minor relationships, the proofs are highly non-constructive in nature. In essence, they demonstrate existence without providing explicit procedures to uncover the relevant obstructions or minor subgraphs.

In many cases, Robertson and Seymour proofs use the probabilistic method or Compactness theorem from logic to show existence of minors, without constructing them. Similarly, the structural decompositions defined are proven to exist, but cannot always be algorithmically obtained. Thus, extracting explicit algorithms remains an ongoing challenge.

Wigderson’s Algebraic Proof – New Connections with Complexity Theory

High-Level Sketch of Wigderson’s Proof Approach

In 2019, theoretical computer scientist Avi Wigderson presented a new algebraic proof of the Graph Minor Theorem based on an analogy with complexity theory. At a high-level, Wigderson establishes a parallel between minor-closed graph families and complexity classes closed under an algebraic “graph minor” operator.

By borrowing concepts from hardness and completeness, he is able to characterize minor relationships algebraically using graph invariants like tree-width. This avoids the set theoretic arguments of Robertson and Seymour, providing a fresh, complexity-theoretic perspective.

Links to Algebraic Complexity Theory

Intriguingly, Wigderson’s proof interpret minors as algebraic operators over graph structures – conceptually similar to operators in algebraic complexity theory. Just as complexity classes can be characterized by complete problems, minor closed graph families can be described by completeness under the minor operator.

These connections suggest deeper characterizations of algorithms and heuristics for extracting minors based on analogies to canonical algorithm design approaches, though much work remains to make this explicit.

Extracting Algorithms from Non-Constructive Proofs?

Obstacles and Barriers

Unfortunately, translating non-constructive proofs in graph minor theory into concrete algorithms faces massive obstacles. While seminal results like Kuratowski’s theorem and the Graph Minor Theorem guarantee existence of forbidden minors or other structures, the proofs offer little guidance in extracting these minors algorithmically.

Challenges include needles in haystacks when searching large graphs, resolving inconsistencies between existential and algorithmic viewpoints, bypassing dependence on the Compactness theorem or probabilistic method, and navigating explosion of possibilities during enumerations.

Partial Progress and Open Questions

There has been some partial progress in recasting specific graph minor theorems algorithmically, often using recursive greedy localization techniques guided by the proofs. Robertson and Seymour have also smoothed some of their arguments to make them more amenable to algorithmization.

However, many open questions remain about decoding the deep structural content in non-constructive minor theories to yield efficient checking, extraction, and obstruction detection algorithms. This represents an exciting area with rich interplay between proof content and algorithm design.

Implications for Algorithm Design

Non-Constructivity as an Obstacle for Algorithmic Applications

The prevalance of non-constructive arguments in graph minor theory often hinders algorithmic applications. Despite the enormous importance of minor relationships in structural and extremal graph theory, as well as networks and circuit layouts, the absence of explicit minor extraction procedures impedes direct utilization.

Overcoming these barriers requires indirect approaches, such as relying exclusively on minor testing algorithms without localization or pursuing graph minor heuristics guided more by experiments than proofs. Thus non-constructivity erects obstacles for applied algorithm designers.

Alternative Proof Approaches More Amenable to Algorithm Design

To facilitate algorithm design, alternative proof approaches may be preferable in graph minor theory. For instance, Wigderson’s algebraic viewpoint better matches algorithmic thinking around expressions, operators, and completeness. Also, proof approaches that eschew the Compactness theorem in favor of explicit recursive constructions may offer more algorithmic traction.

Translation efforts would also benefit from minor theories expressed in terms of structural decompositions and reductions that manifest directly as efficient data structures and checking procedures within algorithms. By working synergistically across logic, graph theory and algorithms, we can harvest deep minor theory advances for applied use.